Poisson theorem of building autonomous integrals for autonomous systems of total differential equations

The autonomous system of total differential equations and corresponding to it Hamiltonian differential system are considered. The analytical relations (the existence of first integrals and partial solutions, fulfillment of conditions of completely solvability) between these differential systems are established. Using these relations, the Poisson theorem of building autonomous first integrals for autonomous system of total differential equations is proved and statements of the existence of additional autonomous first integrals for this system are obtained.