The linear symmetries of Hill's lunar problem

A symmetry of a Hamiltonian system is a symplectic or anti-symplectic involution which leaves the Hamiltonian invariant. For the planar and spatial Hill lunar problem, four resp. eight linear symmetries are well-known. Algebraically, the planar ones form a Klein four-group Z(2) x Z(2) and the spatial ones form the group Z(2) x Z(2) x Z(2). We prove that there are no other linear symmetries. Remarkably, in Hill's system the spatial linear symmetries determine already the planar linear symmetries.