The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Null Killing Vectors and Reductions of the Self-Duality Equations J. Tafel D. Wojcik Vienna, Preprint ESI 487 (1997) September 17, 1997 Supported by Federal Ministry of Science and Research, Austria Available via

2 Null Killing Vectors and Reductions of the Self-Duality Equations J. Tafel and D. Wojcik # Insitute of Theoretical Physics, University of Warsaw, Ho_za 69, Warsaw, Poland, tafel@fuw.edu.pl # Center for Theoretical Physics, Polish Academy of Science, Al. Lotnikow 32/46, Warsaw, Poland, danek@ifpan.edu.pl 1991 MSC: 17B66, 35Q58 PACS: 0210, 1130 Abstract. We nd all null conformal Killing vectors in R 4 with the metric dtdu? dxdy. We classify 1 and 2-dimensional totally null algebras generated by such vectors. Reductions of the self-dual Yang-Mills equations are obtained for gauge elds invariant with respect to these algebras. 1. Introduction Let R 2;2 denote the manifold R 4 with global coordinates x = (t; u; x; y) and the metric ds 2 = dt du? dx dy (1:1) of the signature ++??. The Yang-Mills eld on R 2;2 is represented by a 1-form A = A dx with values in the Lie algebra of a Lie group G. The form A undergoes the following gauge transformations A! A 0 = g?1 Ag + g?1 dg ; (1:2) where g(x ) 2 G. The eld strengths of A are given by F A + [A ; A ]: (1:3) We say that A is self-dual i F is Hodge self-dual with respect to the metric (1.1). For some choice of orientation this condition amounts to the equations F ux = 0 ; F ty = 0 ; F tu + F xy = 0 : (1:4) Equations (1.4) are integrability conditions of the following linear equations [W1,BZ] for a matrix function (; x ), where det( ) 6= 0 and is a parameter, (@ u + A u? (@ y + A y )) = 0 ; (1:5) 1

3 x + A x? (@ t + A t )) = 0 : (1:6) The system of equations (1.4) is an important example of so-called completely integrable equations (see e.g. [AC]). The point symmetry group of (1.4) is the conformal group C(2; 2) (isomorphic to O(3; 3)). Many lower dimensional completely integrable equations, including KdV and NLS equations [MS], can be obtained from (1.4) (or its complexication) by assuming an invariance of A under a subgroup of C(2; 2) (for reviews, see [W2,AC,MW]). Not all reductions of this type are known, mainly because there is no complete classication of subgroups of the conformal group [Wi] (see [KLG] for partial results in the Euclidean case). In this paper we consider algebras of vector elds on R 2;2 generated by one or two (orthogonal) null conformal Killing vectors K i. Such algebras (we call them 'totally null') correspond to particular subgroups of C(2; 2). An invariance of A under one of these subgroups is equivalent to the equation(s) L Ki A = 0 ; (1:7) where L Ki denotes the Lie derivative along K i. We classify all the totally null algebras modulo conformal transformations (Section 2) and we nd reductions of the self-duality equations for gauge elds satisfying (1.7) (Section 3). Some of these reductions are already known [W3,S,T] (see also [St]). The 2-dimensional algebras generated by K 1, K 2 dene at each point of R 2;2 a totally null plane which is either self-dual (then the tensor K 1[ K 2] satises equations (1.4)) or antiself-dual (then K 1[ K 2] satises equations following from (1.4) by the interchange of x and y). In the case of self-dual planes the reduction of the self-dual Yang-Mills equations is singular in the sense that the linear system (1.5), (1.6) does not reduce to a lower dimensional one (one cannot assume an invariance of with respect to K i [T]). We are not able to prove that, in this case, equations (1.4) are trivial (as pde's) for any gauge group, however, there are indications (Section 3) that this is the case. 2. Null conformal Killing vectors The conformal transformations of R 2;2 are generated by translations, rotations, dilations and the special conformal transformation x 0 = s?2 x ; (2:1) where s 2 = tu?xy. Every 1-dimensional subgroup of C(2; 2) generates a conformal Killing vector eld K on R 2;2 which preserves (1.1) up to a factor a(x) L K g = ag : (2:2) It is easy to write down the general expression for K. However, it is not easy to split the set of vector elds K into classes of elds which are related by conformal transformations of coordinates (this problem is equivalent to nding all conjugacy classes of elements of the 2

4 Lie algebra of C(2; 2) [DPWZ]). We show in the following theorem that such classication is quite simple for null Killing vectors, i.e. under the assumption that K = K t + K u + K x + K y ; K t K u? K x K y = 0 : (2:3) Theorem 1. All null conformal Killing vectors in R 2;2 are given by K = (a 1 u + a 2 y + a 3 )[(b 1 u + b 2 x + b 3 )@ u + (b 1 y + b 2 t + b 4 )@ y ] +(a 1 x + a 2 t + a 4 )[(b 1 u + b 2 x + b 3 )@ x + (b 1 y + b 2 t + b 4 )@ t ]; (2:4) where a i ; b i are real constants dened modulo the transformation a i! ca i, b i! c?1 b i, c 2 R. Every vector (2.4) can be put into one of the inequivalent forms K u ; ((2:5) K = a(u@ u + y@ y ) ; a = const ((2:6) by means of a conformal transformation which preserves an orientation. Proof. Any vector satisfying (2.3) can be written as or or K = (@ y t u x ) (2:7) K = K x + K t (2:8) K = K x + K u : (2:9) Expressions (2.8), (2.9) can be obtained from (2.7) (with = 0 or = 0) by simple interchanges of coordinates which preserve (1.1). For this reason we will focus on the case (2.7). In this case the Killing equation (2.2) gives rise to the following x = 0 u () = 0 ; t () = 0 y () = 0 ; x u = 0 y t () = 0 ; y u () = 0 x t = 0 ; t () u x y = 0 : (2:14) For = 0 or = 0 equations (2.10)-(2.14) yield, respectively, K = (b 1 u + b 2 x + b 3 )@ u + (b 1 y + b 2 t + b 4 )@ y ; (2:15) K = (a 1 u + a 2 y + a 3 )@ y + (a 1 x + a 2 t + a 4 )@ t : (2:16) 3

5 Expressions (2.15), (2.16) are special cases of (2.4). If 6= 0 then equation (2.14) follows from equations (2.10)-(2.13). Equations (2.12), (2.13) can be replaced u x = 0 y t = 0 ; y u = 0 t x = 0 (2:18) (for instance, the rst equation in (2.17) follows, in virtue of (2.10), from the rst equation in (2.12)). A direct consequence of (2.17) and (2.18) is that the functions and satisfy the wave y = 0 y = 0 : (2:19) Equations (2.10), (2.11) can be written as the following system of dierential equations for the x log = 0 u log x ; t log x y log t u : (2:21) Integrability conditions of (2.20), (2.21) are linear equations for and, which yield, in virtue of (2.19), = 1 + t 2 + x 3 ; = 1 + u 2 + x 3 ; (2:22) where i = i (u; y) and i = i (t; y). Substituting (2.22) into (2.17) and (2.18) yields = a 1x + a 2 t + a 4 a 1 u + a 2 y + a 3 ; = b 1u + b 2 x + b 3 b 1 y + b 2 t + b 4 ; (2:23) where a i and b i are constants. Given (2.23) one can solve equations (2.20), (2.21) for. The corresponding vector eld (2.7) takes the form (2.4). This ends the proof of the rst part of the theorem. In order to prove that every eld (2.4) can be transformed into (2.5) or (2.6) we will consider separately elds for which exactly n of the constants a 1, a 2, b 1, b 2 vanish. If n = 0 then the rescaling t 0 = a 2 b 2 t, u 0 = a 1 b 1 u, x 0 = a 1 b 2 x, y 0 = a 2 b 1 y transforms all constants a 1, a 2, b 1, b 2 to identity. In the new coordinates K takes the form (here and in what follows we will omit primes after every transformation of coordinates) In virtue of the translation K = (u + y + a 3 )[(u + x + b 3 )@ u + (y + t + b 4 )@ y ] +(x + t + a 4 )[(u + x + b 3 )@ x + (y + t + b 4 )@ t ] : (2:24) t 0 = t + b 4 ; u 0 = u + a 3 ; x 0 = x + b 3? a 3 ; y 0 = y (2:25) followed by the special conformal transformation (2.1) K transforms into K = (1 + au)(@ u ) + (1? ay)(@ t ) ; (2:26) 4

6 where a = a 4. For a 6= 0 a shift in u and y yields K = au(@ u ) + ay(@ y ) ; (2:27) an expression which transforms into (2.6) by means of the transformation t 0 = t+y, u 0 = u, x 0 = x + u, y 0 = y. For a = 0 equation (2.26) yields K u t ; (2:28) hence (2.5) follows by means of the transformation t =?t 0? u 0? x 0? y 0, u =?u 0, x = u 0 + x 0, y = u 0 + y 0. If exactly one of the constants a 1, a 2, b 1, b 2 vanishes we can achieve a 2 = 0, a 1 b 1 b 2 6= 0 by a simple interchange of coordinates. The constants a 1, b 1, b 2 can be set equal to identity due to the transformation t 0 = b 2 t, u 0 = a 1 b 1 u, x 0 = a 1 b 2 x, y 0 = b 1 y. The transformation (2.25) followed by (2.1) yield K = au(@ u ) + (1? ay)(@ t ) ; a = const : (2:29) For a 6= 0 one obtains (2.27) which is equivalent to (2.6). For a = 0 the transformation t =?u 0, u =?t 0? x 0, x = x 0, y = u 0 + y 0 put K into the form (2.5). If exactly two of the constants a 1, a 2, b 1, b 2 vanish we can obtain either or a 1 = a 2 = 0 ; b 1 b 2 6= 0 (2:30) b 1 = a 2 = 0 ; a 1 b 2 6= 0 (2:31) by an interchange of coordinates. In the case (2.30) the transformation t 0 = b 2 t + b 4, u 0 = b 1 u + b 3, x 0 = b 2 x, y 0 = b 1 y leads to K = (x + u)(c u + c x ) + (t + y)(c t + c y ) (2:32) where c 1 = b 1 a 3, c 2 = b 2 a 4. In virtue of the transformation t 0 = t? y, u 0 = x + u, x 0 = x? u, y 0 = t + y one obtains K = (c 2 + c 1 )(u@ u + y@ y ) + (c 2? c 1 )(u@ x + y@ t ) : (2:33) For c 1 6= c 2 an appriopriate rescaling of t and x yields again (2.27). For c 1 = c 2 relation (2.33) coincides with (2.6). For c 2 =?c 1 = c we obtain the expression which transforms to (2.5) under the transformation t = t0 u 0 K = 2c(u@ x + y@ t ) ; (2:34)? x 0 ; u = 1 ; x = 2cu 0 y 0 y 0 5 y 0 ; y = t 0 2cy 0 : (2:35)

7 In the case (2.31) we rst make the transformation t 0 = b 2 t + b 4, u 0 = a 1 u + a 3, x 0 = a 1 (b 2 x + b 3 ), y 0 = y, which leads to K = u(x@ u + t@ y ) + (x + c)(t@ t + x@ x ) ; (2:36) where c = b 2 a 4? b 3 a 1. The vector K takes the form K =?cu@ u + (1? cy)@ y (2:37) upon the transformation (2.1). For c 6= 0 a shift of y yields (2.6). For c = 0 one obtains K y which is equivalent to (2.5). If exactly three of the constants a 1, a 2, b 1, b 2 vanish we can assume that b 1 6= 0, a 1 = a 2 = b 2 = 0. The transformation t 0 = t, u 0 = b 1 u + b 3, x 0 = x, y 0 = b 1 y + b 4 leads to K of the form (2.33), hence (2.5) or (2.6) follows. If a 1 = a 2 = b 1 = b 2 = 0 then K corresponds to a null translation and it can be easily transformed to (2.5). Thus, any eld of the form (2.4) can be transformed to (2.5) or (2.6) by means of a conformal transformation. To show that it can be done by a conformal transformation which preserves an orientation it is sucient to note u is invariant under the transformation t 0 = t ; u 0 = u ; x 0 = y ; y 0 = x (2:38) and that u@ u + y@ y is invariant, modulo the factor -1, under the transformation t 0 = s?2 u ; u 0 = s?2 t ; x 0 = s?2 y ; y 0 = s?2 x (2:39) where s 2 = tu? xy. Transformations (2.38), (2.39) change an orientation of R 2;2. Given a conformal transformation which transforms K into (2.5) or (2.6) one can always combine it with (2.38) or (2.39) in order to obtain a transformation which preserves the orientation. The u and a(u@ u + y@ y ) are not related by any conformal transformation. This can be proved by solving the u 0 = a(u@ u + y@ y ) for a coordinate transformation and showing that such transformation cannot be conformal. Q. E. D. Given the classication of null innitesimal conformal isometries of R 2;2 one can use it to nd 2-dimensional algebras of vector elds (2.4). If these vectors are orthogonal to each other we will call the corresponding algebra a totally null 2-dimensional subalgebra of the conformal algebra. Theorem 2. Every totally null 2-dimensional subalgebra of the conformal algebra of R 2;2 is equivalent, modulo a conformal transformation which preserves an orientation, to one of the following Spanf@ u y g ; Spanf@ u ; x@ u + t@ y g ; Spanf@ u ; u@ u + y@ y g ; (2:40) Spanf@ u x g ; Spanf@ u ; y@ u + t@ x g ; Spanf@ u ; u@ u + x@ x g : (2:41) 6

8 Tangent planes dened by (2.40) and (2.41) are, respectively, self-dual and antiself-dual. Proof. One can assume the following commutation relations for a basis (K 1 ; K 2 ) of a subalgebra under consideration [K 1 ; K 2 ] = K 1 ; = 0; 1 : (2:42) Due to theorem 1 there are cooordinates x such that K 1 is given by (2.5) or (2.6). In these coordinates K 2 takes the general form (2.4). If K 1, K 2 are to dene self-dual totally null planes K 2 has to take the form K 2 = (b 1 u + b 2 x + b 3 )@ u + (b 1 y + b 2 t + b 4 )@ y : (2:43) For K 2 given by (2.43) and K 1 given by (2.6) equation (2.42) cannot be satised. For K 1 u equation (2.42) yields b 1 =. For = b 2 = 0 one obtains the algebra Spanf@ u y g. For = 0, b 2 6= 0 a simple shift in x and t leads to the algebra Spanf@ u ; x@ u + t@ y g. For = 1 one can easily get rid of b 3 and b 4. If b 2 = 0 then the algebra Spanf@ u ; u@ u + y@ y g follows. For b 2 6= 0 one obtains the same algebra due to the transformation t 0 = t, u 0 = u + b 2 x, x 0 = x, y 0 = y + b 2 t. The rst two algebras in (2.40) are abelian and the third one is nonabelian. The abelian ones are nonequivalent since there is no conformal transformation of coordinates such u, x@ u + t@ y are spanned u 0 y 0. The algebras (2.41) are obtained from (2.40) by the transformation t 0 = t, u 0 = u, x 0 = y, y 0 = x which changes an orientation (hence self-dual planes become antiself-dual). Q. E. D. 3. Reductions of the self-duality equations In terms of the gauge potentials A = (A t ; A u ; A x ; A y ) the self-duality equations (1.4) take the u A x A u + [A u ; A x ] = 0 ; t A y A t + [A t ; A y ] = 0 ; t A u A t + [A t ; A u ] x A y A x + [A x ; A y ] = 0 : (3:3) In this section we consider reductions of equations (3.1)-(3.3) for the Yang-Mills elds invariant under one or two-dimensional algebras given in Section 2 (see (2.5), (2.6), (2.40), (2.41)). For all the algebras except (2.40) the reduced equations are related to a linear system which follows from (1.5), (1.6) under the assumption that is anihilated by vector elds from the algebra (this is also true for all other 1 or 2-dimensional subalgebras of the conformal algebra). We say that the Yang-Mills eld A is invariant with respect to an algebra generated by vector elds K i i L Ki A = 0 (3:4) in some gauge. (Note that for the considered algebras condition (3.4) is equivalent to the more general requirement that A is preserved by K i up to gauge transformations [HSV].) 7

9 If K u then it follows from (3.4) that all the functions A are independent of u. Due to (1.2) one can impose the gauge condition Then equation (3.1) yields A x = 0 : (3:5) A u = B(t; y) : (3:6) It follows from (3.2) that there is a function J(t; x; y) 2 G such that Equation (3.3) yields the following condition for J A t = t J ; A y = y J : x y J) + t J; B] t B = 0 : (3:8) Thus, for the symmetry K u, equations (3.1)-(3.3) reduce to (3.8), where B(t; y) can be chosen arbitrarily. For K u and G = SL(2; C) a more sophisticated reduction of the self-duality equations can be obtained as follows [W3,S]. In this case the freedom of gauge transformations (1.2) with g = g(t; y) allows to transform A u into one of the following forms A u = if A u = 0 ; (3:9) 1 0 ; f = f(t; y) ; (3:10) 0?1 0 1 A u = : (3:11) 0 0 In the case (3.9) equations (3.1)-(3.3) reduce to linear ones. In the case (3.10) one obtains the following generalization [Z] of the NLS equation i ;t ;xy + = 0 ; (3:12) where?i ~ ;t ~ ;xy + ~ = 0 ; (3:13) ;y = ( ~ ) ;x : (3:14) In the case (3.11) equations (3.1)-(3.3) reduce either to a system of ordinary and linear equations or to the following generalization of the KdV equation 4 ;ty + ;xyyy? 8 ;y ;xy? 4 ;x ;yy = 0 : (3:15) For K = u@ u + y@ y the symmetry condition (3.4) yields A t = ~ A t ; A u = (uy)? 1 2 ~ Au ; A x = ~ A x ; A y = (uy)?1=2 ~ Ay (3:16) 8

10 where ~ A = ~ A (t; x; z) and z = y=u. By means of a gauge transformation with g = g(t; x; z) one obtains (3.5). Then, from (3.1) it follows x A u = 0. Using again (1.2) (now with some g = g(t; z)) leads to In virtue of (3.16) and (3.17) equations (3.2), (3.3) yield A x = A u = 0 : t ~ Ay? z 1 z ~ At + [ ~ At ; ~ Ay ] = 0 ; (3:18) z 3 z ~ At x ~ Ay = 0 : (3:19) It follows from (3.18) that there is a function J(t; x; z) such that In terms of J equation (3.19) takes the following form A t = t J ; A y = u?1 z J : (3:20) (@ x + z@ t z J) + t J; z J] = 0 : (3:21) Thus, for the symmetry (2.6), the self-duality equations (3.1)-(3.3) reduce to equation (3.21) for a function J(t; x; z) 2 G. It seems that in this case one cannot obtain equations similar to (3.12)-(3.15) (for G = SL(2; C)). The invariance of A with respect to one of the algebras (2.40) yields, respectively, A = ~ A ; (3:22) A t = ~ A t? yt?2 ~ Ay ; A u = t?1 ~ Au ; A x = ~ A x? yt?2 ~ Au ; A y = t?1 ~ Ay ; (3:23) A t = ~ At ; A u = y?1 ~ Au ; A x = ~ Ax ; A y = y?1 ~ Ay ; (3:24) where ~ A = ~ A (t; x). In all the cases the self-duality equations (3.1)-(3.3) reduce to the following x ~ Au + [ ~ Ax ; ~ Au ] = 0 ; t ~ Ay + [ ~ A t ; ~ A y ] = 0 ; t ~ Au + [ ~ A t ; ~ A u ] x ~ Ay + [ ~ A x ; ~ A y ] = 0 : (3:27) In order to simplify equations (3.25)-(3.27) we assume the gauge condition ~A x = 0 (3:28) (note that relations (3.22)-(3.24) are preserved by transformation (1.2) with g = g(t; x)). In this gauge equation (3.25) yields ~A u = B(t) ; (3:29) 9

11 where B is a Lie algebra valued function of t. Equation (3.26) can be solved by introducing a potential J(t; x) such that ~A t = t J ; ~ Ay = J?1 C(x)J ; (3:30) where C is a Lie algebra valued function of x. Equation (3.27) yields t J; B] x (J?1 CJ) t B = 0 : (3:31) Thus, in the case of symmetries (2.40), the self-duality equations (3.1)-(3.3) reduce to equation (3.31) for J(t; x), B(t) and C(x). For these algebras one cannot reduce the linear equations (1.5), (1.6) by assuming K 1 ( ) = K 2 ( ) = 0 since then equation (1.5) would imply the constraints A u = A y = 0 which do not follow from (3.1)-(3.3). Equation (3.31) is related to a pair of Lax operators in a similar way as in the case of completely integrable ordinary equations. For the gauge group SU(2) equations (3.25)-(3.27) can be integrated explicitly as follows. As before, we assume the gauge condition (3.28) and we obtain (3.29). Equations (3.26), (3.27) can be considered as algebraic linear equations for ~ A t, which are solvable i the following equations are satised T r( ~ t ~ Ay ) = 0 ; (3:32) T r( ~ t ~ Au + ~ x ~ Ay ) = 0 ; (3:33) T r( ~ A t ~ Ay ) + T r( ~ A t ~ Au + ~ A x ~ Ay ) = 0 : (3:34) Equations (3.32)-(3.34) are equivalent to the following algebraic conditions for B and ~ A y T r(b) 2 = c 1 t 2 + 2c 2 t + c 3 ; (3:35) T r(b ~ A y ) =?c 1 tx? c 2 x + c 4 t + c 5 ; (3:36) T r( ~ A y ) 2 = c 1 x 2? 2c 4 x + c 6 ; (3:37) where c i are real constants. Thus, the self-duality equations for SU(2) gauge elds with symmetries (2.40) do not reduce to interesting dierential equations. We do not know whether this is also true for other gauge groups however our results (unpublished) for SL(2; C) and SU(3) are also discouraging (we expect similar results for any gauge algebra which admits an ad-invariant scalar product). The invariance of the Yang-Mills eld A with respect to one of the algebras (2.41) yields, respectively, A = ~ A (t; y) ; (3:38) A t = ~ A t? x ~ A x ; A u = t ~ A u ; A x = t ~ A x ; A y = ~ A y? x ~ A u ; (3:39) A t = ~ A t ; A u = x?1 ~ Au ; A x = x?1 ~ Ax ; A y = ~ A y ; (3:40) 10

12 where ~ A = ~ A (t; y). Equations (3.1)-(3.3) take the following form [ ~ Ax ; ~ Au ] = ~ Au ; t ~ y ~ At + [ ~ At ; ~ Ay ] = 0 ; t ~ Au + [ ~ A t ; ~ A u y ~ Ax? [ ~ A y ; ~ A x ] = 0 ; (3:43) where = 0 in the cases (3.28) and (3.39) and = 1 in the case (3.40). It follows from (3.42) that ~A t = ~ Ay = 0 (3:44) in some gauge. From (3.44) and (3.43) one obtains ~A u y R ; ~ Ax t R ; (3:45) where R = R(t; y) is a Lie algebra valued function. It has to satisfy the equation [T] [@ t y R] y R ; (3:46) which follows from (3.41). Thus, for symmetries (2.41), the self-duality equations (3.1)- (3.3) reduce to equation (3.46) for R(t; y), where = 0; 1. For = 1 and a compact gauge group equation (3.46) admits only the trivial solutions R = R(y). For = 0 and G = SU(2) equation (3.46) is also trivial since it is equivalent to R = R(y) or y R + f@ t R = 0, where f is a real function. However, in other cases it might be interesting. For instance, when = 0, G = SU(3) y R is generic in the sense that it has three distinct eigenvalues, equation (3.46) is equivalent to the nonlinear t R = f y R + if 2 ((@ y R) 2? 1 3 T r(@ y R) 2 ) ; (3:47) where f 1, f 2 are real functions which can be arbitrarily prescribed. Equation (3.47) can be replaced by the following quasi linear equation for Q y t Q y (f 1 Q + if 2 (Q 2? 1 3 T rq2 )) : (3:48) Acknowledgements. This work is partially supported by the Erwin Schrodinger Institute for Mathematical Physics and the Polish Committee on Scientic Research (KBN Grant no. 2 P03B 85 09). References [AC] M. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, [BZ] A.A. Belavin and V.E. Zakharov, Yang-Mills equations as inverse scattering problem, Phys. Lett. 73B (1978)

13 [DPWZ] D. Z. Djokovic, J. Patera, P. Winternitz and H. Zassenhaus, Normal forms of elements of classical real and complex Lie and Jordan algebras, J. Math. Phys. 24 (1983) [HSV] J. Harnad, S. Shnider and L. Vinet, Group actions on principal bundles and invariance conditions for gauge elds, J. Math. Phys. 21 (1980) [KLG] M. Kovalyov, M. Legare and L. Gagnon, Reductions by isometries of the self-dual Yang-Mills equations in four-dimensional Euclidean space, J. Math. Phys. 34 (1993) [MS] L.J. Mason and G.A. Sparling, Nonlinear Schr"odinger and Korteweg de Vries are reductions of self-dual Yang-Mills, Phys. Lett. 137A (1989) 29. [MW] L.J. Mason and N.M.J. Woodhouse, Integrability, Self-Duality, and Twistor Theory, Clarendon Press, Oxford [S] J. Schi, Integrability of Chern-Simons-Higgs vortex equations and dimensional reduction of the self-dual Yang-Mills equations to three dimensions in Painleve Transcendents, eds D. Levi and P. Winternitz, Plenum Press, New York [St] I.A.B. Strachan, Null reductions of the Yang-Mills self-duality equations and integrable models in (2+1)-dimensions, in Applications of Analytic and Geometric Methods to Nonlinear Dierential Equations, ed. P.A. Clarkson, NATO ASI series C 413, Kluwer, Dordrecht [T] J. Tafel, Two-dimensional reductions of the self-dual Yang-Mills equations in self-dual spaces, J. Math. Phys. 34 (1993) [W1] R.S. Ward, On self-dual gauge elds, Phys. Lett. 61A (1977) 81. [W2] R.S. Ward, Integrable systems in twistor theory, in Twistors in Mathematics and Physics, eds T.N. Bailey and R.J. Baston, Cambridge University Press, [W3] R.S. Ward, unpublished results. [Wi] P. Winternitz, private communication. [Z] V.E. Zakharov, Inverse scattering method, in Solitons, eds. R.K. Bullough and P.J. Caudrey, Springer-Verlag, Berlin