Small periodic solutions of nonlinear systems of differential equations with constant deviation

We consider a nonlinear system of differential equations in a general case with a singular matrix at the derivatives, with a vector deviation which depends on a parameter. We seek for a periodic solution to the system in the set of trigonometric series such that the sequences of their coefficients belong to the space $l_1$.
We use the method, representing a space as a direct sum of subspaces, and the method of a fixed point of a nonlinear operator as the main investigation techniques.
We reduce the question on the existence of a periodic solution to that of the solvability of an operator equation, whose principal part is defined in a finite-dimensional space.