  In this paper we carry out a detailed analysis of the static spherically
symmetric solutions of a sixth-derivative Lee-Wick gravity model in the
effective delta source approximation. Previous studies of these solutions have
only considered the particular case in which the real and the imaginary part of
the Lee-Wick mass $\mu=a +i b$ are equal. However, as we show here, the
solutions exhibit an interesting structure when the full parameter space is
considered, owing to the oscillations of the metric that depend on the ratio
$b/a$. Such oscillations can generate a rich structure of horizons, a sequence
of mass gaps and the existence of multiple regimes for black hole sizes
(horizon position gaps). In what concerns the thermodynamics of these objects,
the oscillation of the Hawking temperature determines the presence of multiple
mass scales for the remnants of the evaporation process and may permit the
existence of cold black holes with zero Hawking temperature $T$ and
quasi-stable intermediate configurations with $T \approx 0$ and a long
evaporation lifetime. For the sake of generality, we consider two families of
solutions, one with a trivial shift function and the other with a non-trivial
one (dirty black hole). The latter solution has the advantage of reproducing
the modified Newtonian-limit metric of Lee-Wick gravity for small and large
values of $r$.
