A parametrised version of Moser's modifying terms theorem

A sharpened version of Moser's 'modifying terms' KAM theorem is derived, and it is shown how this theorem can be used to investigate the persistence of invariant tori in general situations, including those where some of the Floquet exponents of the invariant torus may vanish. The result is 'structural' and can be applied to dissipative, Hamiltonian, and symmetric vector fields; moreover, we give variants of the result for real analytic, Gevrey regular ultradifferentiable and finitely differentiable vector fields. In the first two cases, the conjugacy constructed in the theorem is shown to be Gevrey smooth in the sense of Whitney on the set of parameters that satisfy a "Diophantine'' non-resonance condition.