  We study the symmetries of the static effective Hamiltonian of a driven
superconducting nonlinear oscillator, the so-called squeeze-driven Kerr
Hamiltonian, and discover a remarkable quasi-spin symmetry $su(2)$ at integer
values of the ratio $\eta=\Delta /K$ of the detuning parameter $\Delta $ to the
Kerr coefficient $K$. We investigate the stability of this newly discovered
symmetry to high-order perturbations arising from the static effective
expansion of the driven Hamiltonian. Our finding may find applications in the
generation and stabilization of states useful for quantum computing. Finally,
we discuss other Hamiltonians with similar properties and within reach of
current technologies.
