  We analyze possibilities to obtain a globally regular stationary
generalization for the ultrastatic wormhole with a repulsive scalar field found
by Bronnikov and by Ellis in 1973. The extreme simplicity of this static
solution suggests that its stationary version could be obtainable analytically
and should be globally regular, but no such generalization has been found. We
analyze the problem and find that the difficulty originates in the vacuum
theory, since the scalar field can be eliminated within the Eris-Gurses
procedure. The problem then reduces to constructing the spinning generalization
for the vacuum wormhole sourced by a thin ring of negative tension. Solving the
vacuum Ernst equations determines the $g_{00}$, $g_{0\varphi}$ metric
components and hence the AMD mass ${\rm M}$ and angular momentum ${\rm J}$, all
of these being specified by the ring source. The scalar field can be included
into consideration afterwords, but this only affects $g_{rr}$ and
$g_{\vartheta\vartheta}$ without changing the rest. Within this approach, we
analyze a number of exact stationary generalizations for the wormhole, but none
of them are satisfactory. However, the perturbative expansion around the static
vacuum background contains only bounded functions and presumably converges to
an exact solution. Including the scalar field screens the singularity at the
ring source and renders the geometry regular. This solution describes a
globally regular spinning wormhole with two asymptotically flat regions. Even
though the source itself is screened and not visible, the memory of it remains
in $g_{00}$ and $g_{0\varphi}$ and accounts for the ${\rm M}\propto {\rm J}^2$
relation typical for a rotating extended source. Describing stationary
spacetimes with an extended source is a complicated problem, which presumably
explains the difficulty in finding the solution.
