| Abstract: Any equivariant irreducible vector bundle for the conformal group $SO_0(4, 1)$ on the 3-sphere $S^3$ is parametrized by an odd number $(= 2N + 1)$ and a complex number $\lambda$. On the other hand,
any equivariant irreducible vector bundle for the conformal group $SO_0(3, 1)$ on the 2-sphere $S^2$ is a line bundle, and is parametrized by an integer number $m$ and a complex number $\nu$. In the present talk, we consider the problems of construction and classification of all differential operators, that are symmetry breaking operators with respect to the conformal pair $SO_0(4, 1) \supset SO_0(3, 1)$, from a vector bundle $V^{2N+1}_{\lambda}$ over the 3-sphere to a line bundle $\mathcal{L}_{m,\nu}$ over the 2-sphere:
$$\mathbb{D}:C^{\infty}(S^3,\mathcal{V}^{2N+1}_{\lambda})\rightarrow C^{\infty}(S^2,\mathcal{L}_{m,\nu})$$ In particular, we solve these problems when the parameters satisfy the condition $|m| \geq N$. |