# Preliminary Group Classification and some exact solutions of 2Hessian equation

## TL;DR

This paper performs a group classification of 3D nonlinear 2-hessian equations, identifying symmetry structures and deriving new invariant models and exact solutions using Lie algebra methods.

## Contribution

It introduces an optimal system of Lie subalgebras for 2-hessian equations and finds new nonlinear invariant models with non-trivial invariance properties.

## Key findings

- Identified an optimal system of Lie subalgebras (A1 to A12) for the equation.
- Derived new nonlinear invariant models with non-trivial invariance.
- Presented exact solutions for the 2-hessian equation.

## Abstract

We study the class of 3-dimensional nonlinear 2-hessian equations mentioned in the text. We perform preliminary group classification on 2-hessian equation. In fact, we find additional equivalence transformation on the space (x,y,z,u,f), with the aid of a method, then we take their projections on the space (x,y,z,f), so we prove an optimal system of one-dimensional Lie sub algebras of this equation is generated by A1 to A12, which introduced in theorem 2, ultimately, A number of new interesting nonlinear invariant models are obtained which have non-trivial in variance algebras. The result of these works is a wide class of equations which summarized in table. So at the end of this work, some exact solutions of 2-hessian equation are presented. The paper is one of the few applications of an algebraic approach to the group classification using Lie method.