Non-integrability of the generalized two fixed centres problem (Q1766798)

The paper presents a generalization of the classical two fixed centres problem in which the standard Newtonian potential is replaced by \(V= - ar^{-2n}\). The proper Hamilton function has the form \[ H= \frac{1}{2}(p_x^2 + p_y^2)-[(x-1)^2+y^2]^{-n}- [(x+1)^2 + y^2]^{-n}, \] where \(n\in\mathbb R\) and \((x,y,p_x,p_y) \) are canonical variables in \(\mathbb R^4\) equipped with the standard symplectic structure. The four cases of integrability of this Hamiltonian system are known: 1) \(n=0\) (the trivial case of a free motion); 2) \(n=1/2\) (classical two centres problem which is separable in elliptic coordinates); 3) \(n=-1\) (the case of two uncoupled harmonic oscillators); 4) \(n=-2\) (natural Hamiltonian system with homogeneous potential of degree four; this case is also separable and integrable in elliptic coordinates). Here, the authors prove the following theorem: the complexified system defined by Hamiltonian (1) admits a meromorphic additional first integral functionally independent of \(H\) if and only if \(n= -2, -1, 0\), or \(1/2\).