Complete integrability of coupled quartic anharmonic oscillators

We complete the study of the Painlevé (P) analysis and integrability of the N coupled quartic anharmonic oscillators defined by $H_N$=1/2 ${\mathcal{J}}_{\mathrm{i}=1\mathrm{}}^{\mathrm{N}}$ ${\mathrm{p}}_{{\mathrm{x}}_{\mathrm{i}}}^2$+ ${\mathcal{J}}_{\mathrm{i}=1\mathrm{}}^{\mathrm{N}}$ ${\mathrm{A}}_{\mathrm{i}}$${\mathrm{x}}_{\mathrm{i}}^2$+ ${\mathcal{J}}_{\mathrm{i}=1\mathrm{}}^{\mathrm{N}}$ ${\mathrm{\alpha }}_{\mathrm{i}}$${\mathrm{x}}_{\mathrm{i}}^4$ +1/2 ${\mathcal{J}}_i$,${\mathit{j}=1\mathrm{}}^{\mathrm{N}}$ ${\mathrm{\beta }}_{\mathrm{ij}}$${\mathrm{x}}_{\mathrm{i}}^2$${\mathrm{x}}_{\mathrm{j}}^2$, i≠j, by listing all the P cases [including those cases which we did not present in Lakshmanan and Sahadevan, Phys. Rev. A 31, 861 (1985)] and deriving the appropriate N involutive integrals for each one of them.