Stationary two-dimensional flow near a free critical point of an incompressible viscoelastic Maxwell medium with upper, lower, and corotational convective derivatives in the rheological constitutive law is considered. Analysis of the analytical unstationary solution found earlier (S. V. Meleshko, N. P. Moshkin, and V. V. Pukhnachev, On exact analytical solutions of equations of Maxwell incompressible viscoelastic medium. Int. J. Non-Lin. Mech., 105:152–157, 2018) provides a new class of stationary solutions. The solutions found comprise both already known as well as substantially new solutions. Nonsingular solutions of the stress tensor at the critical point and bounded at infinity are constructed. Exact analytical formulae for the stress tensor with the Weissenberg number Wi=1/2 are obtained.