Summary: We study integration theory for a vakonomic dynamical equation modeling a nonlinear nonholonomic system. First, the first integrals of the vakonomic equation are given. Secondly, using a cyclic integral and the energy integral respectively, the order of the vakonomic equation is reduced, and a generalized Routh equation and a generalized Whittaker equation are obtained. Third, the canonical equations and the variational equations of vakonomic form are extended; by using a first integral, an integral invariant is constructed. Finally, the basic integral variant and the integral invariants of Poincare-Cartan form and Poincare form for vakonomic dynamics are given.