Is a UCM fluid flow near a stationary point always singular?

Frequently observed divergence of numerical solutions to benchmark flows of the UCM viscoelastic fluid is a known and widely discussed issue. Some authors consider such singularities "invincible". The article argues this position, to which end it considers two typical flows with a stagnation point, often a place of the flows' singularity: counterflows and a flow spread over a wall. For the counterflows numerical and asymptotic analytical solutions are presented. Both kinds of flows turn out regular in the stagnation points, in particular, for high Weissenberg numbers. A good accordance is demonstrated between the analytical and numerical results.