Let $M$ be an odd-dimensional manifold. An *almost contact metric structure* on $M$ is a 4-tuple $(\xi, \eta, \phi, g)$, where $\xi$ is a vector field, $\eta$ a one-form, $\phi$ an endomorphism of the tangent bundle, and $g$ a Riemannian metric, all enjoying the following conditions:

\begin{aligned} \phi^2 &= -I +\xi\otimes \eta\\ \eta(\xi) &= 1\\ g(X,Y) &= g(\phi X, \phi Y) + \eta(X)\eta(Y). \end{aligned}

An almost contact metric structure is a contact metric structure if the following integrability condition is satisfied: \begin{equation} g(X,\phi Y)=d\eta(X,Y). \end{equation}

In CR geometry, contact metric structures appear naturally; if $(M,\eta)$ is a pseudohermitian manifold, then it defines a contact metric structure, with $\xi$ the Reeb vector field, $\phi$ the complex structure on the holomorphic tangent bundle, and $g$ the Webster metric.

Questions: Is there an example of a contact metric structure on some odd-dimensional manifold that does not come from a pseudohermitian structure? Is there a class of manifolds for which pseudohermitian structures are in bijection with contact metric structures?