Differential invariants and group foliation for the complex Monge-Ampere equation

We apply the method of group foliation to the complex Monge-Ampere equation (CMA(2)) with the goal of establishing a regular framework for finding its non-invariant solutions. We employ the infinite symmetry subgroup of the e quation, the group of unimodular biholomorphisms, to produce a foliation of the solution space into leaves which are orbits of solutions with respect to the symmetry group. Accordingly, CMA(2) is split into an automorphic sys tem and a resolvent system which we derive in this paper. This is an intric ate system and here we make no attempt to solve it in order to obtain non-i nvariant solutions. We obtain all differential invariants up to third order for the group of un imodular biholomorphisms and, in particular, all the basis differential inv ariants. We construct the operators of invariant differentiation from which all higher differential invariants can be obtained. Consequently, we are a ble to write down all independent partial differential equations with one r eal unknown and two complex independent variables which keep the same infin ite symmetry subgroup as CMA(2). We prove explicitly that applying operator s of invariant differentiation to third-order invariants we obtain all four th-order invariants. Ar this level we have all the information which is nec essary and sufficient for group foliation. We propose a new approach in the method of group foliation which is based o n the commutator algebra of operators of invariant differentiation. The res olving equations are obtained by applying this algebra to differential inva riants with the status of independent variables. Furthermore, this algebra together with Jacobi identities provides the commutator representation of t he resolvent system. This proves to be the simplest and most natural way of arriving at the resolving equations.