On a Novel Class of Integrable ODEs Related to the Painlevé Equations

One of the authors has recently introduced the concept of conjugate Hamiltonian systems: the solution of the equation $h=H(p,q,t),$ where $H$ is a given Hamiltonian containing $t$ explicitly, yields the function $t=T(p,q,h)$, which defines a new Hamiltonian system with Hamiltonian $T$ and independent variable $h.$ By employing this construction and by using the fact that the classical Painlevé equations are Hamiltonian systems, it is straightforward to associate with each Painlevé equation two new integrable ODEs. Here, we investigate the conjugate Painlevé II equations. In particular, for these novel integrable ODEs, we present a Lax pair formulation, as well as a class of implicit solutions. We also construct conjugate equations associated with Painlevé I and Painlevé IV equations.

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