Transcendental first integrals of dynamical systems on the tangent bundle to the sphere

In this paper, we examine the existence of transcendental first integrals for some classes of systems with symmetries. We obtain sufficient conditions of existence of first integrals of second-order nonautonomous homogeneous systems that are transcendental functions (in the sense of the theory of elementary functions and in the sense of complex analysis) expressed as finite combinations of elementary functions.