Degenerate bifurcation of the rotating patches

In this paper we study the existence of doubly-connected rotating patches for Euler equations when the classical non-degeneracy conditions are not satisfied. We prove the bifurcation of the V-states with two-fold symmetry, however for higher $m-$fold symmetry with $m\geq3$ the bifurcation does not occur. This answers to a problem left open in \cite{H-F-M-V}. Note that, contrary to the known results for simply-connected and doubly-connected cases where the bifurcation is pitchfork, we show that the degenerate bifurcation is actually transcritical. These results are in agreement with the numerical observations recently discussed in \cite{H-F-M-V}. The proofs stem from the local structure of the quadratic form associated to the reduced bifurcation equation.