  Anomalous enstrophy dissipation of incompressible flows in the inviscid limit
is a significant property characterizing two-dimensional turbulence. It
indicates that the investigation of non-smooth incompressible and inviscid
flows contributes to the theoretical understanding of turbulent phenomena. In
the preceding study, a unique global weak solution to the Euler-$\alpha$
equations, which is a regularized Euler equations, for point-vortex initial
data is considered, and thereby it has been shown that, as $\alpha \rightarrow
0$, the evolution of three point vortices converges to a self-similar
collapsing orbit dissipating the enstrophy in the sense of distributions at the
critical time. In the present paper, to elucidate whether or not this singular
orbit can be constructed independently on the regularization method, we
consider a functional generalization of the Euler-$\alpha$ equations, called
the \textit{Euler-Poincar\'{e} models}, in which the incompressible velocity
field is dispersively regularized by a smoothing function. We provide a
sufficient condition for the existence of the singular orbit, which is
applicable to many smoothing functions. As examples, we confirm that the
condition is satisfied with the Gaussian regularization and the vortex-blob
regularization that are both utilized in the numerical scheme solving the Euler
equations. Consequently, the enstrophy dissipation via the collapse of three
point vortices is a generic phenomenon that is not specific to the
Euler-$\alpha$ equations but universal within the Euler-Poincar\'{e} models.
