Darboux transformations and the symmetric fourth Painlevé equation

This paper is concerned with the group symmetries of the fourth Painleve equation P-IV, a second-order nonlinear ordinary differential equation. It is well known that the parameter space of P-IV admits the action of the extended affine Weyl group A(2)((1)). As shown by Noumi and Yamada, the action of A(2)((1)) as Backlund transformations of P-IV provides a derivation of its symmetric form SP4. The dynamical System SP4 is also equivalent to the isomonodromic deformation of an associated three-by-three matrix linear system (Lax pair). The action of the generators of A(2)((1)) on this Lax pair is derived using the Darboux transformation for an associated third-order operator