  In the present work the local form of certain Calabi-Yau metrics possessing a
local Hamiltonian Killing vector is described in terms of a single non linear
equation. The main assumptions are that the complex $(3,0)$-form is of the form
$e^{ik}\widetilde{\Psi}$, where $\widetilde{\Psi}$ is preserved by the Killing
vector, and that the space of the orbits of the Killing vector is, for fixed
value of the momentum map coordinate, a complex 4-manifold, in such a way that
the complex structure of the 4-manifold is part of the complex structure of the
complex 3-fold. The link with the solution generating techniques of [26]-[28]
is made explicit and in particular an example with holonomy exactly SU(3) is
found by use of the linearization of [26], which was found in the context of D6
branes wrapping a holomorphic 1-fold in a hyperkahler manifold. But the main
improvement of the present method, unlike the ones presented in [26]-[28], does
not rely in an initial hyperkahler structure. Additionally the complications
when dealing with non linear operators over the curved hyperkahler space are
avoided by use of this method.
