An infinitely smooth symmetric convex body K ⊂ R d is called k -separably integrable, 1 ≤ k < d , if its k -dimensional isotropic volume function V K , H (t) = H d ({ x ∈ K : dist (x , H ⊥) ≤ t }) can be written as a finite sum of products in which the dependence on H ∈ Gr k (R d) and t ∈ 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. [ABSTRACT FROM AUTHOR]