Communications in Mathematics and Applications Vol. 12, No. 4, pp. 825–833, 2021 ISSN 0975-8607 (online); 0976-5905 (print) Published by RGN Publications http://www.rgnpublications.com DOI: 10.26713/cma.v12i4.1516 Research Article Optimal System and Exact Solutions of Monge-Ampere Equation Tooba Feroze * 1 and Mohsin Umair 2 1 Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad, Pakistan 2 Roots International Schools and Colleges, Kohistan Campus, Wah Cantt., Pakistan Received: February 8, 2021 Accepted: May 28, 2021 Abstract. A solution that remains unchanged when transformed under Lie group of point symmetries of the differential equation is an invariant solution of the differential equation. Optimal system of Lie group of point symmetry generators provide all possible invariant solutions of differential equation. Here, using optimal system of Lie point symmetry generators group invariant solutions are obtained. Using these solutions, exact solutions of non-homogeneous Monge-Ampere equation have been presented here. Keywords. Lie symmetries; Adjoint representations; Commutator relation; Conjugacy classes; Invariant equations Mathematics Subject Classification (2020). 76M60; 22E70; 35A30; 58J70 Copyright © 2021 Tooba Feroze and Mohsin Umair. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The Monge-Ampere equation u xx u yy − u 2 xy + f ( x, y) = 0, (1.1) is a semi-linear non-homogeneous partial differential equation with f ( x, y) as non-homogeneous part of the equation. The name “Monge-Ampere equation” has been derived from its early formulation in two different directions. One by the French mathematician, civil engineer * Email: tferoze@sns.nust.edu.pk