  In this paper, motivated by the classical notion of a Strebel quadratic
differential on a compact Riemann surfaces without boundary we introduce the
notion of a quasi-Strebel structure for a meromorphic differential of an
arbitrary order. It turns out that every differential of even order k exceeding
2 satisfying certain natural conditions at its singular points admits such a
structure. The case of differentials of odd order is quite different and our
existence result involves some arithmetic conditions. We discuss the set of
quasi-Stebel structures associated to a given differential and introduce the
subclass of positive k-differentials. Finally, we provide a family of examples
of positive rational differentials and explain their connection with the
classical Heine-Stieltjes theory of linear differential equations with
polynomial coefficients.
