  $\ell$-boson stars are static, spherical, multi-field self-gravitating
solitons. They are asymptotically flat, finite energy solutions of Einstein's
gravity minimally coupled to an odd number of massive, complex scalar fields. A
previous study assessed the stability of $\ell$-boson stars under spherical
perturbations, finding that there are both stable and unstable branches of
solutions, as for single-field boson stars ($\ell=0$). In this work we probe
the stability of $\ell$-boson stars against non-spherical perturbations by
performing numerical evolutions of the Einstein-Klein-Gordon system, with a 3D
code. For the timescales explored, the $\ell$-boson stars belonging to the
spherical stable branch do not exhibit measurable growing modes. We find,
however, evidence of zero modes; that is, non-spherical perturbations that
neither grow nor decay. This suggests the branching off towards a larger family
of equilibrium solutions: we conjecture that $\ell$-boson stars are the
enhanced isometry point of a larger family of static (and possibly stationary),
non-spherical multi-field self-gravitating solitons.
