Generalized Kazdan-Warner equations associated with a linear action of a torus on a complex vector space

We introduce generalized Kazdan-Warner equations on Riemannian manifolds associated with a linear action of a torus on a complex vector space. We show the existence and the uniqueness of the solution of the equation on any compact Riemannian manifold. As an application, we give a new proof of a theorem of Baraglia [5] which asserts that a cyclic Higgs bundle gives a solution of the periodic Toda equation.