  The Liouvillian exceptional points for a quantum Markovian master equation of
an oscillator in a generic environment are obtained. They occur at the points
when the modified frequency of the oscillator vanishes, whereby the eigenvalues
of the Liouvillian become real. In a generic system there are two parameters
that modify the oscillator's natural frequency. One of the parameters can be
the damping rate. The exceptional point then corresponds to critical damping of
the oscillator. This situation is illustrated by the Caldeira--Leggett (CL)
equation and the Markovian limit of the Hu--Paz--Zhang (HPZ) equation. The
other parameter changes the oscillator's effective mass whereby the exceptional
point is reached in the limit of extremely heavy oscillator. This situation is
illustrated by a modified form of the Kossakowski--Lindblad (KL) equation. The
eigenfunctions coalesce at the exceptional points and break into subspaces
labelled by a natural number $N$. In each of the $N$-subspace, there is a
$(N+1)$-fold degeneracy and the Liouvillian has a Jordan block structure of
order-$(N+1)$. We obtain the explicit form of the generalized eigenvectors for
a few Liouvillians. Because of the degeneracies, there is a freedom of choice
in the generalized eigenfunctions. This freedom manifests itself as an
invariance in the Jordan block structure under a similarity transformation
whose form is obtained. We compare the relaxation of the first excited state of
an oscillator in the underdamped region, critically damped region which
corresponds to the exceptional point, and overdamped region using the
generalized eigenvectors of the CL equation.
