  On a Riemann surface with a holomorphic $r$-differential, one can naturally
define a Toda equation and a cyclic Higgs bundle with a grading. A solution of
the Toda equation is equivalent to a harmonic metric of the Higgs bundle for
which the grading is orthogonal. Here we focus on a general non-compact Riemann
surface with an $r$-differential which is not necessarily meromorphic at
infinity. We introduce the notion of complete solution of the Toda equation,
and we prove the existence and uniqueness of a complete solution by using
techniques for both Toda equations and harmonic bundles. Moreover, we show some
quantitative estimates of the complete solution.
