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The symmetry group structures of two dimensional coupled nonlinearShr\"{o}dinger equations are considered. We first show that the equations admitinfinite dimensional symmetry algebra as well as the corresponding symmetrygroup depending on four arbitrary functions of one variable. Then we show somephysical symmetries and an affine loop algebra contained in the symmetryalgebra of the equations. Third, we give the complete classifications of finitedimension (less than four) subalgebras of the symmetry algebra under theadjoint group of the symmetry group. These results provide the theoretical andcomputational basis for the further study of the equations with symmetrymethods.

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