I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well known fact that in a neighbourhood $U$ of the zero section of the cotangent bundle $T^{*}M$, $M$ admits a complexification. By Stenzel and Guillemin this complexification is called the adapted complex structure. It is even Kähler. In this complexification, the phase function $\varphi$ solves the homogenous Monge-Ampere equation. My question is now: are there Kähler structures on a neighbourhood $U$ of the zero section in the cotangent bundle of $M$, which turn $M$ into a Lagrangian submanifold and such that $M$ is isometrically embedded in $U$, BUT differ from the adapted complex structure (up to biholomorphism)? Do all such structures come from an adapted complex structure? Is this already known or can it be derived easily?