This paper systematically investigates the Lie symmetry analysis of a class of 3-dimensional non-linear 2-Hessian equations \(u_{xx}u_{yy}+u_{xx}u_{yy}+u_{yy}u_{zz}=u_{xy}^2+u_{yz}^2+u_{xz}^2+f\), where f is an arbitrary smooth function f of the variables (x, y, z). In fact, the preliminary group classification of the 2-Hessian equation was carried out. That means, we find one-dimensional Lie symmetry extensions of the principal 4-dimensional sub-algebra of the equivalence algebra of these equations. So, we find additional equivalence transformation on the space (x, y, z, f), with the aid of Bila’s method, and, we take their projections on this space. Then we obtain an optimal system of one-dimensional Lie sub-algebras of these equations which are generated by some vectors and presented on Theorem (5.1). Some new non-linear invariant models are obtained which have non-trivial invariance algebras. The results of the preliminary group classification are some inequivalent equations which summarized in a table. Finally, some exact solutions of the \(2-\)Hessian equation are presented, and some figures for the obtained solutions are depicted.

本文系统地研究了一类3维非线性2-Hessian方程\（u_ {xx} u_ {yy} + u_ {xx} u_ {yy} + u_ {yy} u_ {zz}的Lie对称性分析= u_ {xy} ^ 2 + u_ {yz} ^ 2 + u_ {xz} ^ 2 + f \），其中f是变量（x，  y，  z）的任意平滑函数f。实际上，已经对2-Hessian方程进行了初步的组分类。这意味着，我们找到这些方程的等价代数的主要4维子代数的一维Lie对称扩展。因此，我们在空间（x，  y，  z，  f），借助Bila的方法，然后我们将其投影到该空间上。然后，我们获得了这些方程的一维李子代数的最佳系统，该系统由一些矢量生成并在定理（5.1）中给出。获得了具有非平凡不变代数的一些新的非线性不变模型。初步组分类的结果是一些不等式，总结在表中。最后，给出了\（2- \） Hessian方程的一些精确解，并描述了所获得解的一些图。