The Manakov system and Fuchsian equations

In our work we consider the Manakov system and the monodromy matrix associated with it, which plays a key role in the construction of multiphase solutions. All integrable nonlinear equations from the hierarchy of the Manakov system are expressed in terms of the elements of this monodromy matrix. The spectral curves of the multiphase solutions of each of the equations from this hierarchy are determined by the characteristic equation of the monodromy matrix. Stationary equations, which are satisfied by multiphase solutions to the evolutionary hierarchy equations, can also be written using elements of the monodromy matrix. In the present paper, the simplest nontrivial stationary equations are considered and solutions are constructed. These solutions are expressed in terms of integrals from solutions to the Fuchsian equations. In dependence on the values of the parameters of the stationary equations, these Fuchsian equations can have five, four or three singular points. For each solutions that are expressed in terms of elliptic or elementary functions, the parameter values are found. In the case of Fuchsian equations with three singular points (hypergeometric equations), the solutions to the Manakov system have the form of positons. The coefficients of the corresponding spectral curves equations are found for all the considered solutions. In cases where solutions to the Manakov system can be classified as positons, non-hyperelliptic spectral curves have twofold branching points along with simple ones. This fact has a direct correspondence with the properties of the spectral curves of positons of those integrable nonlinear equations whose spectral curves are hyperelliptic.