Enumeration problems associated with Zolotarev fractions

Known properties of the Chebyshev polynomials: simple critical points and only two (finite) critical values. Those properties uniquely determine the polynomials modulo affine transformations of dependent and independent variables.
A similar property of Zolotarev fractions: simple critical points and only four critical values ​​generates already many classes of rational functions modulo projective transformations of the dependent and independent variables. They will be listed in the talk.