  A class of metrizable vector bundles in the general framework of generalized
Lie algebroids have been presented in the eight reference. Using a generalized
Lie algebroid we obtain the Lie algebroid generalized tangent bundle of a
vector bundle. This Lie algebroid is a new example of metrizable vector bundle.
A new class of Lagrange spaces, called by use, generalized Lagrange
(\rho?;\eta?)-space, Lagrange (\rho?;\eta?)-space and Finsler
(\rho?;\eta?)-space are presented. In the particular case of Lie algebroids,
new and important results are presented. In particular, if all morphisms are
identities morphisms, then the classical results are obtained.
