In some instances, the reference system can be chosen in such a way that it is an analytically solvable problem. This is particularly true, for example, in path integral molecular dynamics, a technique for studying quantum mechanical systems at finite temperature.
As a paradigm for this case, we shall consider a particular realization
of 
in the problem considered in the last section. Consider
the problem described by the dynamics
in which is replaced by a harmonic force 
.
Here, the reference system Liouville operator will be
which is just the Liouville operator for a harmonic oscillator, a
problem for which we know the analytical solution. Thus, the
action of the reference system propagator 
on the phase space vector (x,p) is
The correction Liouville operator will still be given by
Now, if the same factorization is made, i.e.
the action of the operator can be evaluated in closed form. Note that
Thus, acting in (x,p) gives
which can be simplified to read
It is straightforward to see that the integrator is both symplectic and reversible.
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