Three-dimensional analysis of asymptotic forms of the solutions to the sixth Painlevé equation.

The purpose of this work is to clarify the question: can solutions to the sixth Painlevé equation have asymptotic forms of the Boutroux type or other asymptotic forms different from the forms obtained earlier by methods of plane power geometry. Methods of space power geometry are used to solve this task. For that, the sixth Painlevé equation is reduced to a system of two differential equations. We find asymptotic forms of solutions to the system for $x \to 0,1,\infty$. As we found out, the sixth Painlevé equation has 12 new families of the Boutroux type asymptotic forms different from forms found earlier by methods of plane power geometry.