Reflecting function preserving polinomial perturbations of Abel equation

In this paper the method of constructing of Abel differential equations possessing the same Mironenko reflecting function is studied. For another equation with the same reflecting function to be constructed, the necessary and sufficient conditions for given Abel equation $\dot x = a_0 (t) + a_1(t)x + a_2 (t)x^2 + a_3 (t)x^3$ are achieved.