  We analyze the symmetries in an open quantum system composed by three coupled
and detuned harmonic oscillators in the presence of a common heat bath. It is
shown analytically how to engineer the couplings and frequencies of the system
so as to have several degrees of freedom unaffected by decoherence,
irrespective of the specific spectral density or initial state of the bath.
This partial thermalization allows observing asymptotic entanglement at
moderate temperatures, even in the nonresonant case. This latter feature cannot
be seen in the simpler situation of only two oscillators, highlighting the
richer structural variety of the three-body case. When departing from the
strict conditions for partial thermalization, a hierarchical structure of
dissipation rates for the normal modes is observed, leading to a long transient
where quantum correlations such as the quantum discord are largely preserved,
as well as to synchronous dynamics of the oscillators quadratures.
