On the nonintegrability of the generalized van der Waals Hamiltonian system (Q2738123)

The generalized van der Waals Hamiltonian in cylindrical coordinates \(\rho,\lambda,z\) is NEWLINE\[NEWLINEH=\frac{1}{2} \left( P^2 + \frac{\Lambda^2}{\rho^2}+Z^2 \right) -\frac{1}{(\rho^2+z^2)^{1/2}}+\frac{\gamma}{2}(\rho^2+\beta^2z^2),NEWLINE\]NEWLINE where \(P,\Lambda,Z\) are the conjugate momenta, \(\gamma\) is a fixed constant and \(\beta\) is a parameter. Since \(\lambda\) is ignorable, \(\Lambda\) is a second integral for \(H\). In the cases \(\beta=1,2,\frac{1}{2}\), a third integral of motion, in involution with \(H,\Lambda\) is known to exist and the system is integrable in the Liouville sense. By applying the nonintegrability theorem of \textit{J. J. Morales-Ruiz} and \textit{J. P. Ramis} [Methods Appl. Anal. 8, 33-95, 97-111 (2001; Zbl 1140.37352)], the authors prove that, for all other values of \(\beta\), the Hamiltonian \(H\) does not possess a third, independent real integral in involution, which has a meromorphic extension to \(C^6\).