An integrable deformation of the well-known Neumann-Rosochatius system is studied by considering generalised bosonic spinning solutions on the eta-deformed AdS(5) x S-5 background. For this integrable model we construct a 4x4 Lax representation and a set of integrals of motion that ensures its Liouville integrability. These integrals of motion correspond to the deformed analogues of the Neumann-Rosochatius integrals and generalise the previously found integrals for the.-deformed Neumann and (AdS(5) x S-5)(eta) geodesic systems. Finally, we briefly comment on consistent truncations of this model.