Systems of two nonlinear ordinary differential equations of the first order admitting nonlinear superpositions are investigated using Lie's enumeration of groups on the plane. It is shown that the systems associated with two-dimensional Vessiot-Guldberg-Lie algebras can be integrated by quadrature upon introducing Lie's canonical variables. The knowledge of a symmetry group of a system in question is not needed in this approach. The systems associated with three-dimensional Vessiot-Guldberg-Lie algebras are classified into 13 standard forms 10 of which are integrable by quadratures and three are reduced to Riccati equations.