We discuss the equivalent form of the Lévy-Leblond equation such that the nilpotent matrices are two-dimensional.We show that this equation can be obtained in the non-relativistic limit of the (2+1)-dimensional Dirac equation.Furthermore,we analyze the case with four-dimensional matrices,propose a Hamiltonian for the equation in (3+1) dimensions,and solve it for a Coulomb potential.The quantized energy levels for the hydrogen atom are obtained,and the result is consistent with the non-relativistic quantum mechanics.