  We argue that the $\kappa$-deformation is related to a factorization of a Lie
group, therefore {\em an approproate version of $\kappa$-Poincar\'{e} does
exist on the $C^*$-algebraic level}. The explict form of this factorization is
computed that leads to an ``action'' of the Lorentz group (with space
reflections) considered in Doubly Special Relativity theory. The orbit
structure is found and ``the momentum manifold'' is extended in a way that
removes singularities of the ``action'' and results in a true action. Some
global properties of this manifold are investigated
