  Two new classes of metrizable vector bundles have been presented in the
papers [1] and [4]. The Lie algebroid generalized tangent bundle of a dual
vector bundle is presented. This Lie algebroid is a new example of metrizable
vector bundle. A new class of Hamilton spaces, called by use, generalized
Hamilton (\rho,\eta)-space, Hamilton (\rho,\eta)-space and Cartan
(\rho,\eta)-space are presented. The results obtained in the particular case of
Lie algebroids emphasize the importance and the utility of our new method by
work. In particular, if all morphisms are identities morphisms, then the
classical results are obtained.
