  We study the dissipative preparation of pure non-Gaussian states of a target
mode which is coupled both linearly and quadratically to an auxiliary damped
mode. We show that any pure state achieved independently of the initial
condition is either (i) a cubic phase state, namely a state given by the action
of a non-Gaussian (cubic) unitary on a squeezed vacuum or (ii) a (squeezed and
displaced) finite superposition of Fock states. Which of the two states is
realized depends on whether the transformation induced by the engineered
reservoir on the target mode is canonical (i) or not (ii). We discuss how to
prepare these states in an optomechanical cavity driven with multiple control
lasers, by tuning the relative strengths and phases of the drives. Relevant
examples in (ii) include the stabilization of mechanical Schr\"odinger cat-like
states or Fock-like states of any order. Our analysis is entirely analytical,
it extends reservoir engineering to the non-Gaussian regime and enables the
preparation of novel mechanical states with negative Wigner function.
