  We consider distributions on $\R^n\setminus{0}$ which satisfy a given set of
partial differential equations and provide criteria for the existence of
extensions to $\R^n$ that satisfy the same set of equations on $\R^n$. We use
the results to construct distributions satisfying specific renormalisation
conditions in the Epstein and Glaser approach to perturbative quantum field
theory. Contrary to other approaches, we provide a unified apporach to treat
Lorentz covariance, invariance under global gauge group and almost homogeneity,
as well as discrete symmetries. We show that all such symmetries can be
recovered by applying a linear map defined for all degrees of divergence. Using
similar techniques, we find a relation between on-shell and off-shell
time-ordered products involving higher derivatives of the fields.
