  We begin by discussing various ways autoequivalences and stability conditions
associated to triangulated categories can interact. Once an appropriate
definition of compatibility is formulated, we derive a sufficiency criterion
for this compatibility. We next apply this criterion to derived categories
associated to Galois covers of the Weierstrass nodal cubic, known as n-gons and
denoted by E_n. These are singular non-irreducible genus 1 curves naturally
arising in variety of contexts, including as certain degenerations of elliptic
curves.
  In particular, fixing the stability condition to be the natural extension of
classical slope to E_n, we explicitly compute the moduli space of stable
objects and its compactification (given by S-equivalence). The compactification
of stable objects with a fixed slope is isomorphic to a disjoint union of E_m
and Z/nZ where m|n; m varies as the slope varies and all such m occur. This
computation is made possible by explicitly constructing the group of all
autoequivalences compatible with the choice of stability condition. It is found
that this group is an extension of the modular group \Gamma_0(n) by a direct
product of Aut(E_n), Pic^0(E_n), and Z.
