The nonhomogeneous Monge–Ampère equation, read as $w_{xx}{w}_{yy}-w_{xy}^2+h\left(w\right)=0$, is a nonlinear equation involving mixed second derivatives with respect to the spatial variables x and y, along with an additional source function $h\left(w\right)$. This equation is observed in several fields, including differential geometry, fluid dynamics, and magnetohydrodynamics. In this study, the Lie symmetry method is used to obtain a detailed classification of this equation. Symmetry analysis leads to a comprehensive classification of the equation, resulting in specific forms of the smooth source function $h\left(w\right)$. Furthermore, the one-dimensional optimal system of the associated Lie algebras is derived, allowing for symmetry reductions that yield several exact invariant solutions of the Monge–Ampère equation. In addition, conservation laws are constructed using the Noether approach, a highly effective and widely used method for deriving conserved quantities. These conservation laws can help evaluate the accuracy and reliability of numerical methods.

This study presents the complete analysis of a (2 + 1)-dimensional nonlinear wave-type partial differential equation with anisotropic power-law nonlinearities and a general power-law source term, which arises in physical domains such as fluid dynamics, nonlinear acoustics, and wave propagation in elastic media, yet their symmetry properties and exact solution structures remain largely unexplored for arbitrary nonlinearity exponents. To fill this gap, a complete Lie symmetry classification of the equation is performed for arbitrary values of m and n, providing all admissible symmetry generators. These generators are then employed to systematically reduce the PDE to ordinary differential equations, enabling the construction of exact analytical solutions. Traveling wave and soliton solutions are derived using Jacobi elliptic function and sine-cosine methods, revealing rich nonlinear dynamics and wave patterns under anisotropic conditions. Additionally, conservation laws associated with variational symmetries are obtained via Noether’s theorem, yielding invariant physical quantities such as energy-like integrals. The results extend the existing literature by providing, for the first time, a full symmetry classification for arbitrary m and n, new families of soliton and traveling wave solutions in multidimensional settings, and associated conserved quantities. The findings contribute both computationally and theoretically to the study of nonlinear wave phenomena in multidimensional cases, extending the catalog of exact solutions and conserved dynamics of a broad class of nonlinear partial differential equations.