Integrable nonconservative dynamical systems on the tangent bundle of the multidimensional sphere

We construct a class of nonconservative systems of differential equations on the tangent bundle of the sphere of any finite dimension. This class has a complete set of first integrals, which can be expressed as finite combinations of elementary functions. Most of these first integrals consist of transcendental functions of their phase variables. Here the property of being transcendental is understood in the sense of the theory of functions of the complex variable in which transcendental functions are functions with essentially singular points.