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An extensive investigation, both numerical and analytical, is performed foran anisotropic extension of the ferromagnetic spin-1 biquadratic model. Theground state phase diagram accommodates three symmetry-protected trivialphases, three coexisting fractal phases and six Luttinger liquid phases. Anovel universality class arises from an instability of a Luttinger liquidtowards an exotic quantum state of matter with infinitely degenerate groundstates. The latter in turn is a scale-invariant quantum state of matter, whichmay be attributed to the coexistence of ${\rm SU}(2)$ spontaneous symmetrybreaking with one type-B Goldstone mode on the characteristic line: $J_y=J_z$,and ${\rm U}(1)$ spontaneous symmetry breaking without any gapless Goldstonemode on the characteristic line $J_x/J_z=0$, together with their cyclicpermutations with respect to $x$, $y$ and $z$.

Instability of the Luttinger liquids towards an exotic quantum state of  matter with highly degenerate ground states: an anisotropic extension of the  ferromagnetic spin-1 biquadratic model