  The many-body localized (MBL) phase is characterized by a complete set of
quasi-local integrals of motion and area-law entanglement of excited
eigenstates. We study the effect of non-Abelian continuous symmetries on MBL,
considering the case of $SU(2)$ symmetric disordered spin chains. The $SU(2)$
symmetry imposes strong constraints on the entanglement structure of the
eigenstates, precluding conventional MBL. We construct a fixed-point
Hamiltonian, which realizes a non-ergodic (but non-MBL) phase characterized by
eigenstates having logarithmic scaling of entanglement with the system size, as
well as an incomplete set of quasi-local integrals of motion. We study the
response of such a phase to local symmetric perturbations, finding that even
weak perturbations induce multi-spin resonances. We conclude that the
non-ergodic phase is generally unstable and that $SU(2)$ symmetry implies
thermalization. The approach introduced in this work can be used to study
dynamics in disordered systems with non-Abelian symmetries, and provides a
starting point for searching non-ergodic phases beyond conventional MBL.
