Power expansions of solutions to the sixth Painlevé equation

By means of Power Geometry, shortly described in § 1, in the generic case we compute all power expansions of solutions to the sixth Painlevé equation at points $x=0$, $x=\infty$ (§ 2) and $x=1$ (§ 3). Three symmetries of the equation allow reducing all these expansions to three basic families. One of them begins by the term with arbitrary power exponent that means a new type of singularity of the equation.