  An infinitely smooth symmetric convex body $K\subset\mathbb R^d$ is called
$k$-separably integrable, $1\leq k<d$, if its $k$-dimensional isotropic volume
function $V_{K,H}(t)=\mathcal H^d(\{\boldsymbol x\in
K:\mathrm{dist}(\boldsymbol x,H^\perp)\leq t\})$ can be written as a finite sum
of products in which the dependence on $H\in\mathrm{Gr}(k,\mathbb R^d)$ and
$t\in\mathbb R$ is separated. In this paper, we will obtain a complete
classification of such bodies. Namely, we will prove that if $d-k$ is even,
then $K$ is an ellipsoid, and if $d-k$ is odd, then $K$ is a Euclidean ball.
This generalizes the recent classification of polynomially integrable convex
bodies in the symmetric case.
