Extension of the Diedron Group for the 4th Painleve Equation by Means of a Power Transformation
The purpose of this article is to expand the transformation dihedral discrete group for the 4th Painlevé equation and find a way to solve the equations of the extended orbit of the 4th Painlevé equation. The following tasks were completed: a power transformation was found that is closed in the class of second-order polynomial ordinary differential equations (ODE); the transformation discrete dihedral group of 12th order was extended using the found power transformation to the 84th order pseudogroup of transformations; the graph of this pseudogroup of transformations was constructed; the found transformation pseudogroup was applied to the 4th Painlevé equation; the equations of the extended orbit of the 4th Painlevé equation were found; a way of solving all 84 equations of the extended orbit of the 4th Painlevé equation through the 4th Painlevé transcendent was indicated. The author of the article managed to find another (power) transformation, closed in the class of fractional polynomial ODEs of the second order, along with two generators of the transformation dihedral group, due to which it was possible to expand the transformation dihedral group of the 4th Painlevé equation. Moreover, the resulting power transformation has one arbitrary parameter, therefore, 72 equations generated by this transformation also have an additional parameter, along with two arbitrary coefficients of the 4th Painlevé equation. The paper uses a method for constructing discrete groups and pseudogroups of transformations, a method for constructing graphs of the found discrete groups and pseudogroups of transformations, the method of “multiplication” of solvable equations by the found discrete pseudogroup of transformations.