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A class of metrizable vector bundles in the general framework of generalizedLie algebroids have been presented in the eight reference. Using a generalizedLie algebroid we obtain the Lie algebroid generalized tangent bundle of avector 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 areidentities morphisms, then the classical results are obtained.

Metrizability of the Lie algebroid generalized tangent bundle and (generalized) Lagrange $(\rho,\eta)$-spaces