  Asteroid (31) Euphrosyne is one of the biggest objects in the asteroid main
belt and the Euphrosyne family occupies a highly inclined region in the outer
main belt and contains a remarkably large number of members, which is
interpreted as an outcome of a disruptive cratering event. The goals of this
adaptive-optics imaging study were threefold: to characterize the shape of
Euphrosyne, to constrain its density, and to search for the large craters that
may be associated with the family formation event. We obtained disk-resolved
images of Euphrosyne using SPHERE/ZIMPOL at ESO's 8.2-m VLT as part of our
large program (ID: 199.C-0074, PI: Vernazza). We reconstructed its 3D-shape
using the adam shape modeling algorithm based on the SPHERE images and the
available lightcurves of this asteroid. We analyzed the dynamics of the
satellite with the genoid meta-heuristic algorithm. Finally, we studied the
shape of Euphrosyne using hydrostatic equilibrium models. Our SPHERE
observations show that Euphrosyne has a nearly spherical shape with the
sphericity index of 0.9888 and its surface lacks large impact craters.
Euphrosyne's diameter is 268+/-6 km, making it one of the top 10 largest main
belt asteroids. We detected a satellite of Euphrosyne -- S/2019 (31) 1-- that
is about 4 km across, on an circular orbit. The mass determined from the orbit
of the satellite together with the volume computed from the shape model imply a
density of 1665+/-242 kg/m^3, suggesting that Euphrosyne probably contain a
large fraction of water ice in its interior. We find that the spherical shape
of Euphrosyne is a result of the reaccumulation process following the impact,
as in the case of (10) Hygiea. However, our shape analysis reveals that,
contrary to Hygiea, the axis ratios of Euphrosyne significantly differ from the
ones suggested by fluid hydrostatic equilibrium following reaccumulation.
