  For any Lagrangean K\"ahler submanifold $M \subset T^*{\Bbb C}^n$, there
exists a canonical hyper K\"ahler metric on $T^*M$. A K\"ahler potential for
this metric is given by the generalized Calabi Ansatz of the theoretical
physicists Cecotti, Ferrara and Girardello. This correspondence provides a
method for the construction of (pseudo) hyper K\"ahler manifolds with large
automorphism group. Using it, a class of pseudo hyper K\"ahler manifolds of
complex signature $(2,2n)$ is constructed. For any hyper K\"ahler manifold $N$
in this class a group of automorphisms with a codimension one orbit on $N$ is
specified. Finally, it is shown that the bundle of intermediate Jacobians over
the moduli space of gauged Calabi Yau 3-folds admits a natural pseudo hyper
K\"ahler metric of complex signature $(2,2n)$.
