  We define a nondegenerate Monge-Amp\`ere structure on a 6-dimensional
manifold as a pair $(\Omega,\omega)$, such that $\Omega$ is a symplectic form
and $\omega$ is a 3-differential form which satisfies $\omega\wedge\Omega=0$
and which is nondegenerate in the sense of Hitchin. We associate with such a
pair a generalized almost (pseudo) Calabi-Yau structure and we study its
integrability from the point of view of Monge-Amp\`ere operators theory. The
result we prove appears as an analogue of Lychagin and Roubtsov theorem on
integrability of the almost complex or almost product structure associated with
an elliptic or hyperbolic Monge-Amp\`ere equation in the dimension 4. We study
from this point of view the example of the Stenzel metric on the cotangent
bundle of the sphere $S^3$.
