  This paper is the second part of our study on the Toda equations and the
cyclic Higgs bundles associated to $r$-differentials over non-compact Riemann
surfaces. We classify all the solutions up to boundedness around the isolated
singularity of an $r$-differential under the assumption that the
$r$-differential is meromorphic or has some type of essential singularity. As a
result, for example, we classify all the solutions on ${\mathbb C}$ if the
$r$-differential is a finite sum of the exponential of polynomials.
