A class of integrable two-dimensional (2D) classical systems with integrals of motion of fourth order in momenta is obtained from the quantum analogues with the help of deformed SUSY algebra. With similar technique a new class of potentials connected with the Lax method is found which provides the integrability of corresponding 2D Hamiltonian systems. In addition, some integrable 2D systems with potentials expressed in elliptic functions are explored.