  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.
