The finite dimensional subalgebra classification of infinite dimensional symmetry algebra of two dimensional coupled nonlinear Schr\"{o}dinger equations.

The symmetry group structures of two dimensional coupled nonlinear Shr\"{o}dinger equations are considered. We first show that the equations admit infinite dimensional symmetry algebra as well as the corresponding symmetry group depending on four arbitrary functions of one variable. Then we show some physical symmetries and an affine loop algebra contained in the symmetry algebra of the equations. Third, we give the complete classifications of finite dimension (less than four) subalgebras of the symmetry algebra under the adjoint group of the symmetry group. These results provide the theoretical and computational basis for the further study of the equations with symmetry methods.