This work presents a symmetry analysis of the nonhomogeneous Monge–Ampère equation \(\varvec{\det D^{2} u = x^{p} y^{q}}\) in \(\varvec{\Omega \subset } \pmb {\mathbb {R}}^{\textbf{2}}\) , with \(\varvec{p, q \in } \pmb {\mathbb {R}}\) . Using Lie group theory, we classify all point symmetries, obtaining a five-dimensional Lie algebra in the generic case, which is solvable and isomorphic to \({\mathbb {R}}^3 \ltimes {\mathbb {R}}^2\) . We prove the equation is nonlinearly self-adjoint and apply Ibragimov’s theorem to derive explicit conservation laws for each symmetry generator. The adjoint representation of the Lie algebra is constructed, and a one-dimensional optimal system of subalgebras is determined, enabling classification of all distinct symmetry reductions. These reductions yield nonlinear ordinary differential equations, analyzed qualitatively via Poincaré compactification to identify critical points at infinity and their stability. The results provide a comprehensive structural description of the equation’s symmetries, conservation laws, and invariant solutions, enhancing understanding of its geometric and analytical properties.
