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Motivated by the definition of rigid centres for planar differential systems, we introduce the study of rigid centres on the center manifolds of differential systems on 
$\mathbb {R}^{3}$. On the plane, these centres have been extensively studied and several interesting results have been obtained. We present results that characterize the rigid systems on 
$\mathbb {R}^{3}$ and solve the centre-focus problem for several families of rigid systems.
We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systems
where h, K, Ψ and ξ are analytic functions defined in the neighbourhood of O with K(O) ≠ 0 or Ψ(O) ≠ 0 and n ≥ 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.
