  Constant mean curvature surfaces in $S^3$ can be studied via their associated
family of flat connections. In the case of tori this approach has led to a deep
understanding of the moduli space of all CMC tori. For compact CMC surfaces of
higher genus the theory is far more involved due to the non abelian nature of
their fundamental group. In this paper we extend the spectral curve theory for
tori developed in \cite{Hi, PiSt} and for genus $2$ surfaces \cite{He3} to CMC
surfaces in $S^3$ of genus $g=k\cdot l$ with commuting $\mathbb Z_{k+1}$ and
$\mathbb Z_{l+1}$ symmetries. We determine their associated family of flat
connections via certain flat line bundle connections parametrized by the
spectral curve. We generalize the flow on spectral data introduced in
\cite{HeHeSch} and prove the short time existence of this flow for certain
families of initial surfaces. In this way we obtain various families of new CMC
surfaces of higher genus with prescribed branch points and prescribed umbilics.
