On a method for constructing algebro-geometric solutions to the zero curvature equation

Under special conditions that hold for a number of applications, we suggest a construction that reduces the calculation of algebro-geometric solutions of the zero curvature equation for $2\times2$ matrices to solving the Jacobi inversion problem on a hyperelliptic Riemann surface and the Riccati equation. An application to the system of equations of the principal chiral field is considered.