  Breaking of time-reversal and point-group spatial symmetries can have a
profound impact on superconductivity. One of the most extraordinary effects,
due to the application of a magnetic field, is represented by the Abrikosov
vortices with charged supercurrents circulating around their cores. Whether a
similar phenomenon can be obtained by exploiting spatial symmetry breaking,
e.g. through electric fields or mechanical strain, is a fundamentally relevant
but not yet fully settled problem. Here, we show that in two-dimensional
spin-singlet superconductors with unusually low degree of spatial symmetry
content, vortices with supercurrents carrying angular momentum around the core
can form and be energetically stable. The vortex has zero net magnetic flux
since it is made up of counter-propagating Cooper pairs with opposite orbital
moments. By solving self-consistently the Bogoliubov - de Gennes equations in
real space, we demonstrate that the orbital vortex is stable and we unveil the
spatial distribution of the superconducting order parameter around its core.
The resulting amplitude has a characteristic pattern with a pronounced angular
anisotropy that deviates from the profile of conventional magnetic vortices.
These hallmarks guide predictions and proposals for the experimental detection.
