  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.
