  In this paper, we address for the 2D Euler equations the existence of rigid
time periodic solutions close to stationary radial vortices of type
$f_0(|x|){\bf 1}_{\mathbb{D}}(x)$, with $\mathbb{D}$ the unit disc and $f_0$
being a strictly monotonic profile with constant sign. We distinguish two
scenarios according to the sign of the profile: defocusing and focusing. In the
first regime, we have scarcity of the bifurcating curves associated with lower
symmetry. However in the focusing case we get a countable family of bifurcating
solutions associated with large symmetry. The approach developed in this work
is new and flexible, and the explicit expression of the radial profile is no
longer required as in [41] with the quadratic shape. The alternative for that
is a refined study of the associated spectral problem based on Sturm-Liouville
differential equation with a variable potential that changes the sign depending
on the shape of the profile and the location of the time period. Deep hidden
structure on positive definiteness of some intermediate integral operators are
also discovered and used in a crucial way. Notice that a special study will be
performed for the linear problem associated with the first mode founded on
Pr\"{u}fer transformation and Kneser's Theorem on the non-oscillation
phenomenon.
