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Suppose that $M \subset Z$ is a compact Lagrangian submanifold of a Kaehler manifold $Z$ with Kaehler form $\omega$. Take a function $f$ so that $\partial \bar\partial f=0$ at every point of $M$ and so that $\partial \bar\partial f \ge 0$ near $M$. Then for a small enough neighborhood of $M$ in $Z$, $\omega+i\partial\bar\partial f$ is a Kaehler form in which $M$ is a Lagrangian manifold with the same induced metric. In your case, take $Z=T^*M$ with the Stenzel metric.