Lax pairs, Backlund transformations and special solutions for ordinary differential equations

The authors investigate a modification of the Weiss-Tabor-Carnevale procedure that enables one to obtain Lax pairs and Backlund transformations for systems of ordinary differential equations. This method can yield both auto-Backlund transformations and, where necessary, Backlund transformations between different equations. In the latter case they investigate the circumstances under which the general Backlund transformations reduce to auto-Backlunds. In addition, special solution families for the second and fourth Painleve transcendents are obtained.