  Discretizations of the Euler top sharing the integrals of motion with the
continuous time system are studied. Those of them which are also Poisson with
respect to the invariant Poisson bracket of the Euler top are characterized.
For all these Poisson discretizations a solution in terms of elliptic functions
is found, allowing a direct comparison with the continuous time case. We
demonstrate that the Veselov--Moser discretization also belongs to our family,
and apply our methods to this particular example.
