  We formulate matrix string models on a class of exact string backgrounds with
non constant RR-flux parameterized by a holomorphic prepotential function and
with manifest (2,2) supersymmetry. This lifts these string theories to M-theory
exposing the non perturbative string interaction which is studied by
generalizing the instanton asymptotic expansion, well established in the flat
background case, to this more general case. We obtain also a companion matrix
model with four manifest supersymmetries in eleven dimensions.
