  The main result of this paper is a construction of finitely additive measures
for higher rank abelian actions on Heisenberg nilmanifolds. Under a full
measure set of Diophantine conditions for the generators of the action, we
construct \emph{Bufetov functionals} on rectangles on $(2g+1)$-dimensional
Heisenberg manifolds. We prove that deviation of the ergodic integral of higher
rank actions is described by the asymptotic of Bufetov functionals for a
sufficiently smooth function. As a corollary, the distribution of normalized
ergodic integrals which have variance 1, converges along certain subsequences
to a non-degenerate compactly supported measure on the real line.
