  A bar-joint framework $(G,p)$ is the combination of a finite simple graph
$G=(V,E)$ and a placement $p:V\rightarrow \mathbb{R}^d$. The framework is rigid
if the only edge-length preserving continuous deformations of the vertices
arise from isometries of the space. This article combines two recent extensions
of the generic theory of rigid and flexible graphs by considering symmetric
frameworks in $\mathbb{R}^3$ restricted to move on a surface. In particular
necessary combinatorial conditions are given for a symmetric framework on the
cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any
finite point group symmetry. In every case when the symmetry group is cyclic,
which we prove restricts the group to being inversion, half-turn or reflection
symmetry, these conditions are then shown to be sufficient under suitable
genericity assumptions, giving precise combinatorial descriptions of symmetric
isostatic graphs in these contexts.
