  Kerr nonlinear oscillators driven by a two-photon process are promising
systems to encode quantum information and to ensure a hardware-efficient
scaling towards fault-tolerant quantum computation. In this paper, we show that
an extra control parameter, the detuning of the two-photon drive with respect
to the oscillator resonance, plays a crucial role in the properties of the
defined qubit. At specific values of this detuning, we benefit from strong
symmetries in the system, leading to multiple degeneracies in the spectrum of
the effective confinement Hamiltonian. Overall, these degeneracies lead to a
stronger suppression of bit-flip errors. We also study the combination of such
Hamiltonian confinement with colored dissipation to suppress leakage outside of
the bosonic code space. We show that the additional degeneracies allow us to
perform fast and high-fidelity gates while preserving a strong suppression of
bit-flip errors.
