  We construct the vector field associated with the GKLS generator for systems
described by Gaussian states. This vector field is defined on the dual space of
the algebra of operators, restricted to operators quadratic in position and
momentum. It is shown that the GKLS dynamics accepts a decomposition principle,
that is, this vector field can be decomposed in three parts, a conservative
Hamiltonian component, a gradient-like and a Choi-Krauss vector field. The last
two terms are considered a "perturbation" associated with dissipation. Examples
are presented for a harmonic oscillator with different dissipation terms.
