Homogeneous generalized functions with respect to one-parametric group

We give the full description of homogeneous generalized functions along the trajectories of arbitrary one-parametric multiplicative group of linear transformations whose generator matrix has eigenvalues with positive real parts. We also study the problem of extension of such functionals from the space of test functions vanishing at the origin up to the whole space $S(\mathbb{R}^n)$, and discuss the conditions of uniqueness of such extension.