  In this paper, an explicit hierarchy of differential equations for the
$\tau$-functions defining the moduli space of curves with automorphisms as a
subscheme of the Sato Grassmannian is obtained. The Schottky problem for
Riemann surfaces with automorphisms consists of characterizing those p.p.a.v.
that are Jacobian varieties of a curve with a non-trivial automorphism. A
characterization in terms of hierarchies of p.d.e. for theta functions is also
given.
