Contact and CR Examples What is an example of a manifold such that:
(A) It is both a contact manifold and a CR manifold
(B) It is a contact manifold but not a CR manifold
(C) It is not a contact manifold but not a CR manifold?
 A: For a contact metric manifold $M$ we observe that $J = \varphi_{\vert D}$, i.e. the restriction
of $\varphi$ to the contact distribution, defines an almost complex structure on $D=\ker\eta$. Then the
associated almost CR-structure of $M$ is given by the holomorphic subbundle
$$H = \{X − i JX\mid X \in D\}$$
of the complexification $TM^{\Bbb C}$ of the tangent bundle $TM$. We say that the associated
almost CR-structure is integrable if $[H,H] \subset H$. This is equivalent to $[J, J ](X, Y ) =
0$, for any $X, Y \in D$, where $[J, J ]$ denotes the Nijenhuis torsion of $J$. It is known that
the associated CR-structure of a three dimensional contact metric manifold is always
integrable (S. Tanno 1989).
A contact metric manifold $M$ such that its associated almost CR-structure is integrable will be referred to as a contact
strongly pseudo-convex CR-manifold.
It should be noted that every
$(\kappa,\mu)$-manifold is a contact strongly pseudo convex integrable CR-manifold (D. E. Blair et al Contact metric manifolds satisfying a nullity condition doi:10.1007/BF02761646).
And answer of your questions:
A) By above comments, every
$(\kappa,\mu)$-manifold  is both a contact manifold and a CR manifold.
B) In Mitric [1991] and Tanno [1992] it was
shown that the tangent sphere bundle with its standard contact metric
structure is a CR-manifold if and only if the base manifold is of constant
curvature. Thus if we choose $M$ with non-constant sectional curvature then the tangent sphere bundle $T_1M$ with its standard contact metric
structure is a contact metric manifold that not a CR-manifold.
For (C) I do not have an example in mind.
A: The way I understand the questions, these may provide possible examples:
(A) These are precisely CR structures of CR codimension 1 with nondegenerate Levi-form. That is, the nondegeneracy of the Levi form implies the maximal nonintegrability of the corresponding hyperplane distribution.
(B) Take any example that is both contact and CR and perturb the CR structure to a non-integrable almost CR structure (any generic perturbation in dimension higher than 3).
(C) Take any hyperplane distribution that is not maximally non-integrable, then it is not contact. Then add any compatible non-integrable almost CR structure, which is locally always possible. 
