  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 which asserts that a cyclic Higgs bundle gives a solution of the
periodic Toda equation.
