Legendre duality on hypersurfaces in Kähler manifolds

In this paper we will give a sufficient condition on real strictly Levi-convex hypersurfaces $M$, embedded in four-dimensional K\"{a}hler manifolds $V$, such that Legendre duality can be performed. We consider the contact form $\theta$ on $M$ whose kernel is the restriction of the holomorphic tangent space of $V$: we will show that if there exists a Legendrian Killing vector field $v$, then the dual form $\beta(\cdot):=d\theta(v,\cdot)$ is a contact form on $M$ with the same orientation than $\theta$.