  We compute the anomalous dimension of the single current operator in the case
of single and doubly deformed asymmetric $\lambda$-models with a general
deformation matrix. Our method uses the underlying geometry of the coupling
space, as well as an auxiliary group interaction, which completely decouples
from the asymmetric model in a specific limit, consistent with the
Renormalization Group flow. Our results are valid to all orders in the
deformation parameters and leading order to the levels of the underlying
current algebras. We specialize our general result to several models of
particular interest that have been constructed in the literature and for which
these anomalous dimensions were not known.
