We consider a four-dimensional PDE possessing partner symmetries mainly on the example of complex Monge-Ampère equation (CMA). We use simultaneously two pairs of symmetries related by a recursion relation, which are mutually complex conjugate for CMA. For both pairs of partner symmetries, using Lie equations, we introduce explicitly group parameters as additional variables, replacing symmetry characteristics and their complex conjugates by derivatives of the unknown with respect to group parameters. We study the resulting system of six equations in the eight-dimensional space, that includes CMA, four equations of the recursion between partner symmetries and one integrability condition of this system. We use point symmetries of this extended system for performing its symmetry reduction with respect to group parameters that facilitates solving the extended system. This procedure does not imply a reduction in the number of physical variables and hence we end up with orbits of non-invariant solutions of CMA, generated by one partner symmetry, not used in the reduction. These solutions are determined by six linear equations with constant coefficients in the five-dimensional space which are obtained by a three-dimensional Legendre transformation of the reduced extended system. We present algebraic and exponential examples of such solutions that govern Legendre-transformed Ricci-flat Kähler metrics with no Killing vectors. A similar procedure is briefly outlined for Husain equation.
Explicit Flow Equations and Recursion Operator of the ncKP hierarchy
导出了非对易Kadomtsev-Petviashvili（ncKP）族流方程的显式表达式.与KP族的流方程相比，我们的结果表明ncKP族的流方程中的附加项确实由动力学坐标\{ $u_i$ \}的分解子组成.给出了流动方程在$n$ -约化下的递推算子。此外，根据2-减少，我们计算一个非局部递归算子$Φ(2)$的非交换Korteweg-de Vries（ncKdV）的层次结构，它产生一个层次结构的本地，高阶流。因此，我们解决了由P. J. Olver和V. V. Sokolov（Commun.Math. Phys.193（1998），245-268）提出的公开问题。
