  We examine some recently-constructed families of asymptotically-AdS$_3
\times$S$^3$ supergravity solutions that have the same charges and mass as
supersymmetric D1-D5-P black holes, but that cap off smoothly with no horizon.
These solutions, known as superstrata, are quite complicated, however we show
that, for an infinite family of solutions, the null geodesic problem is
completely integrable, due to the existence of a non-trivial conformal Killing
tensor that provides a quadratic conservation law for null geodesics. This
implies that the massless scalar wave equation is separable. For another
infinite family of solutions, we find that there is a non-trivial conformal
Killing tensor only when the left-moving angular momentum of the massless
scalar is zero. We also show that, for both these families, the metric degrees
of freedom have the form they would take if they arose from a consistent
truncation on S$^3$ down to a (2+1)-dimensional space-time. We discuss some of
the broader consequences of these special properties for the physics of these
black-hole microstate geometries.
