On the twistor space of a (co-)CR quaternionic manifold

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We characterise, in the setting of the Kodaira-Spencer deformation theory, the twistor spaces of (co-)CR quaternionic manifolds. As an application, we prove that, locally, the leaf space of any nowhere zero quaternionic vector field on a quaternionic manifold is endowed with a natural co-CR quaternionic structure. Also, for any positive integers k and l, with kl even, we obtain the geometric objects whose twistorial counterparts are complex manifolds endowed with a conjugation without fixed points and which preserves an embedded Riemann sphere whose normal bundle is l times the line bundle of Chern number k. We apply these results to prove the existence of natural classes of co-CR quaternionic manifolds.