  The so-called ``adiabatic elimination'' of fast decaying degrees of freedom
in open quantum systems can be performed with a series expansion in the
timescale separation. The associated computations are significantly more
difficult when the remaining degrees of freedom (center manifold) follow fast
unitary dynamics instead of just being slow. This paper highlights how a
formulation with Sylvester's equation and with adjoint dynamics leads to
systematic, explicit expressions at high orders for settings of physical
interest.
