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We consider a new type of ${\cal N}=4$, $d=1$ semi-dynamical multiplet basedon the non-linear version of the mirror multiplet ${\bf (3, 4, 1)}$, with thetriplet of bosonic physical fields parametrizing a three-dimensional sphere$S^3$ of the radius $R$. The limit $R\to\infty$ amounts to the contraction$S^3\to\mathbb{R}^3$ and leads to the linear mirror multiplet ${\bf (3, 4,1)}$. Spin degrees of freedom described by a Wess-Zumino action specify atwo-dimensional surface embedded in the sphere $S^3$. A pair of the examplesconsidered correspond to the round and squashed ``fuzzy'' 2-spheres. We couplethe squashed 2-sphere model to the dynamical mirror multiplet ${\bf (2, 4,2)}$. A notable feature of this coupling is the dependence of the squashingparameter on the bosonic fields $z, \bar z$ of the chiral multiplet.

Non-linear (3, 4, 1) multiplet of ${\cal N} = 4$, $d = 1$ supersymmetry as a semi-dynamical spin multiplet