to the free nonlocal anyonic fields within a path integral approach is accompanied by the appearance of the complicated Jacobian of the transformation [@sreedpr], and this means a nontrivial relic of the statistical gauge field in the theory. Therefore, the Chern-Simons gauge field approach does not give a minimal description of (2+1)-dimensional anyons.

Another, less developed way consists in attempting to describe anyons within the group-theoretical approach analogously to the case of integer and half-integer spin fields, without using Chern-Simons U(1) gauge field constructions. The program of this approach [@ply2]–[@pld] consists in constructing equations for (2+1)-dimensional fractional spin field, subsequent identifying corresponding field action and, finally, in realizing a quantization of the theory to reveal a fractional statistics. Within this approach, there are, in turn, two related possibilities: to use many-valued representations of the (2+1)-dimensional Lorentz group SO(2,1), or to work with the infinite-dimensional unitary representations of its universal covering group, $\overline{\rm SO(2,1)}$ (or $\overline{\rm
SL(2,R)}$, isomorphic to it). Up to now, it is not clear how to construct the action functionals corresponding to the equations for the fractional spin field carrying many-valued representations of SO(2,1) [@ply2; @for; @ply1]. On the other hand, different variants of the equations and some 