Center varieties of complex and real two-dimensional autonomous polynomial differential systems
In the present article we consider the interaction between the solution of the problem of obtaining necessary and sufficient conditions of existence of a time-independent first integral analytical in the neighborhood of the origin for the complex system of the form x = y + P(x, y), y = -x + Q(x, y), where P(x, y), Q(x, y) are polynomials without constant and linear terms, and the solution of the center-focus problem for the real system of the same form.