We consider the sixth Painlev'e equation in the case a,b ≠ 0. By the methods of Power Geometry, near the singular points x=0 and x=∞, we have found all power, power-logarithmic and complicated expansions of its solutions. Near x=0 we have obtained 15 families of expansions, sixth of them are complicated. Using a symmetry of the equation, near x=∞ we have obtained again 15 families of expansions, including 6 complicated.

Mathematical problems and theory of numerical methods