Exotic expansions of solutions to the sixth Painlevé equation

To the sixth Painlevé equation near three its singular points for various values of its four complex parameters, we study asymptotic expansions of its solutions in series of complex powers of the independent variable with constant coefficients, which contain infinitely many terms with fixed real part of the power exponent (they were named as exotic expansions). We show that the series can correspond to solutions with very complicated singularities. At first we compute basic families of exotic expansions (alltogether 9 families), among them 8 families found for the first time, and then by means of symmetries of the sixth Painleve equation, we obtain the rest of families of exotic expansions of the sixth Painleve equation (alltogether with basic families 54 families).