She classical local solvability of the periodic boundary-value problem and Cauchy problem for the system $$ \frac{\partial\Delta\Psi}{\partial t}+\frac{\partial}{\partial x_2}(\Psi_{x_1}\Delta\Psi)-\frac{\partial}{\partial x_1}(\Psi_{x_2}\Delta\Psi)-\Delta^2\omega=F, \Psi=\alpha\frac{\partial\omega}{\partial t}+\beta\omega+\int^t_0S(t-\tau)\omega(\tau)d\tau, \alpha>0, $$ is proved. The system describes two-dimensional motions of Maxwell fluids of order $L=1,2,\dots$ . Bibl. – 6.