Abstract <p>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 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. The stationary equations, which are satisfied by multiphase solutions of the evolutionary hierarchy equations, can also be written using elements of the monodromy matrix. In the present work, the simplest nontrivial stationary equations are considered and solutions are constructed. These solutions are expressed in terms of integrals from solutions to Fuchsian equations. Depending on the values of parameters of the stationary equations, these Fuchsian equations can have five, four, or three singular points. The values of the parameters at which the solutions are expressed through elliptic or elementary functions 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 equations of the corresponding spectral curves are found for all the considered solutions. In the cases where the solutions to the Manakov system can be classified as positons, nonhyperelliptic spectral curves have both twofold branching points and simple points. 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.</p>