  In this paper we use the language of real meromorphic differentials from the
theory of Klein surfaces to describe the metastable states of multiply
connected planar ferromagnetic nanoelements which minimize the exchange energy
and have no side magnetic charges. Those solutions still have enough internal
degrees of freedom which may serve as the Ritz parameters for minimization of
further relevant energy terms or as the dynamical variables for the adiabatic
approach. The nontrivial topology of the magnet itself brings us to several
effects first described for the annulus and observed in the experiment. We
explain the topological constraints on the numbers of vortexes and antivortexes
in the magnet, as well as the algebraic constraints on their positions which
stem from the Abel's theorem. The use of multivalued Prym differentials bring
us to new meron configurations which were not considered in the seminal work of
D.J.Gross.
