planar physical phenomena: fractional quantum Hall effect and high-$T_{c}$ superconductivity [@appl]. Anyons also attract a great attention due to their relationship to the different theoretical fields of research such as conformal field theories and braid groups (see, e.g., refs. [@fub]–[@ler]).

From the field-theoretical point of view, such particles can be described in two, possibly related, ways. The first way consists in organizing a statistical interaction of the scalar or fermionic field with the Chern-Simons U(1) gauge field, that changes spin and statistics of the matter field [@sem]. In this approach, manifestly gauge-invariant nonlocal field operators, constructed from the initial gauge noninvariant matter fields following the line integral prescription of Schwinger, carry fractional spin and satisfy anyonic permutation relations [@sred]. Such redefined matter fields are given on the one-dimensional path — unobservable ‘string’ going to the space infinity and attached to the point in which the initial matter field is given. Therefore in this approach the anyonic field operators are path-dependent and multivalued [@sred] (see also ref. [@fro1]). The initial Lagrangian for the statistically charged matter field can be rewritten in the decoupled form in terms of the anyonic gauge-invariant matter fields and Chern-Simons gauge field. However, such a formal transition 