A two-dimensional unsteady stagnation-point flow of an incompressible viscoelastic fluid is studied theoretically assuming that the fluid obeys the upper convected Maxwell model. To achieve better understanding of the main properties of the governing equations, the system of non-linear equations is transformed to Lagrangian variables. As a result, a closed system of equations of the mixed elliptic–hyperbolic type is obtained. These equations are decomposed into a hyperbolic submodel and a quadrature. The hyperbolic part is responsible for the transport of nonlinear transverse waves in an incompressible Maxwell medium. The system of equations guarantees the existence of the energy integral, which allows one to analyze discontinuous solutions to these equations. It is demonstrated that solutions with strong discontinuities are impossible, though a solution with weak discontinuities can exist. Several numerical examples of the problems of practical interest show that perturbations induced by weak discontinuities in the initial data propagate with a finite speed, which confirms the hyperbolic character of the system.