  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.

High Energy Physics - Theory;Differential Geometry