The first integrals of the equation of motion by using elements of nonlinear functional analysis was reported. The equation 1 can be an ordinary differential equation, a partial differential equation, an integrodifferential equation, or a system of such equations. The theory of transformations is used to establish the invariance of Hamiltonian actions and the invariance of the equations of motion and to find first integrals of these equations. The first integrals of the equations of motion for infinite-dimensional systems acquire numerous applications. various equations and theorems was also given in support of the experiment.