To date, it is not known if smooth initial conditions of the Euler equations with finite energy do or do not blow up in finite time. It is shown that under the assumption of spatial anisotropy, an axisymmetric incompressible and inviscid flow potentially presents a finite-time singularity. The singular flow consists of a quadrupolar structure for the vorticity together with a tangential discontinuity of the swirl velocity. On the discontinuity plane, the velocity field becomes a multivalued singularity. This singularity appears to be generic and robust for a wide number of finite energy initial conditions.