On polynomial integrals and branching of solutions to invertible dynamic systems on the sphere (Q1358169)

\textit{V. V. Kozlov} [Math. Notes 44, No. 1/2, 543-545 (1988); translation from Mat. Zametki 44, No. 1, 100-104 (1988; Zbl 0825.58031)] considered the relationship between polynomial integrals and branching of solutions of a dynamical system describing the motion of a point on the torus \(T^n\) with a plane metric in the field of periodic forcing with meromorphic components. The goal of this note is to extend the results of Kozlov to the dual problem of motion of a point on the \(n\)-sphere \(S^n\subset \mathbb{R}^{n+1}\) with a forcing whose components are meromorphic functions in \(\mathbb{C}^{n+1}\).