On integrable Ermakov-Painlevé IV systems (Q1746662)

The nonlinear coupled systems of Ermakov-Ray-Reid type is the following system \[ \frac{d^2\phi}{dz^2}+\Theta(z)\phi=\frac{1}{\phi^2\psi}\Phi(\psi/\phi), \] \[ \frac{d^2\psi}{dz^2}+\Theta(z)\psi=\frac{1}{\psi^2\phi}\Phi(\phi/\psi). \] This system has a distinguished integral of motion \[ I=1/2(\phi\frac{d\psi}{dz}-\psi\frac{d\phi}{dz})^2+\int^{u=\psi/\phi}\Phi(u)du+\int^{u=\phi/\psi}\Psi(u)du. \] Recently in [\textit{C. Rogers}, Stud. Appl. Math. 133, No. 2, 214--231 (2014; Zbl 1297.35235); ``Hybrid Ermakov-Painlevé-IV systems'', J. Nonlinear Math. Phys. 21, 628--642 (2014)], ``the prototype integrable hybrid Ermakov-Painlevé II and Ermakov-Painlevé IV-type systems were derived via wave packet symmetry reductions of two classes of physically important resonant nonlinear Schrödinger systems.'' In the present paper, the ``novel hybrid Ermakov-Painlevé-IV systems are introduced and an associated Ermakov invariant is used in establishing their integrability.''