Exact solutions and Bäcklund transformations for an extended second Painlevé equation

We consider an extended version of the second Painlevé equation \((\mathrm P_{\mathrm{II}})\), which appears as the simplest member of a recently-derived extended second Painlevé hierarchy. For this third-order system we consider the application of the Ablowitz–Ramani–Segur algorithm, use its auto-Bäcklund transformations ( auto-BTs) to construct sequences of rational solutions and solutions defined in terms of Bessel functions, the latter constituting the analogues for the extended \(\mathrm P_{\mathrm{II}}\) of the well-known Airy function solutions of \(\mathrm P_{\mathrm{II}}\). In addition, we present two new Bäcklund transformations, which extend the Schwarzian \(\mathrm P_{\mathrm{II}}\) equation due to Weiss and an auto-BT due to Gambier. Finally, we use the auto-BTs of extended \(\mathrm P_{\mathrm{II}}\) also to derive a new third-order discrete system.