Conditional symmetry and new classical solutions of the Yang–Mills equations

are based on the ansätze for the Yang–Mills field Aμ(x) suggested by Wu and Yang, Rosen, ’t Hooft, Corrigan and Fairlie, Wilczek, Witten (see [1] and references therein). There were further developments for the self-dual YMEs (which form the first-order system of nonlinear partial differential equations such that system (1) is its differential consequence). Let us mention the Atiyah–Hitchin–Drinfeld–Manin method for obtaining instanton solutions [2] and its generalization due to Nahm. However, the solution set of the self-dual YMEs is only a subset of solutions of YMEs (1) and the problem of construction of new non self-dual solutions of system (1) is, in fact, completely open (see, also [1]). As the development of new approaches to the construction of exact solutions of YMEs is a very interesting mathematical problem, it may also be of importance for physics. The reason is that all famous mathematical models of elementary particles such as solitons, instantons, merons are quite simply particular solutions of some nonlinear partial differential equations. A natural approach to construction of particular solutions of YMEs (1) is to utilize their symmetry properties in the way as it is done in [9, 10, 16] (see, also [15], where the reduction of the Euclidean self-dual YMEs is considered). The apparatus of the theory of Lie transformation groups makes it possible to reduce system of partial differential equations (PDEs) (1) to systems of nonlinear ordinary differential equations (ODEs) by using special ansätze (invariant solutions) [10, 18, 20]. If one succeeds in constructing general or particular solutions of the said ODEs (which is an extremely difficult problem), then on substituting the results in the corresponding ansätze one gets exact solutions of the initial system of PDEs (1). Another possibility of construction of exact solutions of YMEs is to use their conditional (non-Lie) symmetry (for more details about conditional symmetry of equations of mathematical physics, see [6, 8] and also [10, 12]) which has much in common with