  We study one-mode Gaussian quantum channels in continuous-variable systems by
performing a black-box characterization using complete positivity and trace
preserving conditions, and report the existence of two subsets that do not have
a functional Gaussian form. Our study covers as particular limit the case of
singular channels, thus connecting our results with their known classification
scheme based on canonical forms. Our full characterization of Gaussian channels
without Gaussian functional form is completed by showing how Gaussian states
are transformed under these operations, and by deriving the conditions for the
existence of master equations for the non-singular cases. We show that although
every functional form can be found in the vicinity of the identity, one of them
does not parametrize unitary channels.
