  The paper is devoted to differential geometry of singular distributions
(i.e., of varying dimension) on a Riemannian manifold. Such distributions are
defined as images of the tangent bundle under smooth endomorphisms. We prove
the novel divergence theorem with the divergence type operator and deduce the
Codazzi equation for a pair of singular distributions. Tracing our Codazzi
equation yields expression of the mixed scalar curvature through invariants of
distributions, which provides some splitting results. Applying our divergence
theorem, we get the integral formula, generalizing the known one, with the
mixed scalar curvature of a pair of transverse singular distributions.
