Group Foliation Approach to the Complex Monge–Ampére Equation

We apply the group foliation method to find noninvariant solutions of the complex Monge–Ampére equation $(\textrm{CMA}_2)$. We use the infinite symmetry subgroup of the $\textrm{CMA}_2$ to foliate the solution space into orbits of solutions with respect to this group and correspondingly split the $\textrm{CMA}_2$ into an automorphic system and a resolvent system. We propose a new approach to group foliation based on the commutator algebra of operators of invariant differentiation. This algebra together with Jacobi identities provides the commutator representation of the resolvent system. For solving the resolvent system, we propose symmetry reduction, which allows deriving reduced resolving equations.