  The Euclidean cone metrics coming from q-differentials on a closed surface of
genus g > 1 define an equivalence relation on homotopy classes of closed curves
declaring two to be equivalent if they have the equal length in every such
metric. We prove an analog of the result of Randol for hyperbolic metrics
(building on the work of Horowitz): for every integer q > 1, the corresponding
equivalence relation has arbitrarily large equivalence classes. In addition, we
describe how these equivalence relations are related to each other.
