  The separability of the Hamilton-Jacobi equation has a well-known connection
to the existence of Killing vectors and rank-two Killing tensors. This paper
combines this connection with the detailed knowledge of the compactification
metrics of consistent truncations on spheres. The fact that both the inverse
metric of such compactifications, as well as the rank-two Killing tensors can
be written in terms of bilinears of Killing vectors on the underlying "round
metric," enables us to perform a detailed analyses of the separability of the
Hamilton-Jacobi equation for consistent truncations. We introduce the idea of a
separating isometry and show that when a consistent truncation, without
reduction gauge vectors, has such an isometry, then the Hamilton-Jacobi
equation is always separable. When gauge vectors are present, the gauge group
is required to be an abelian subgroup of the separating isometry to not impede
separability. We classify the separating isometries for consistent truncations
on spheres, $S^n$, for $n=2, \dots, 7$, and exhibit all the corresponding
Killing tensors. These results may be of practical use in both identifying when
supergravity solutions belong to consistent truncations and generating
separable solutions amenable to scalar probe calculations. Finally, while our
primary focus is the Hamilton-Jacobi equation, we also make some remarks about
separability of the wave equation.
