Construction and classification of differential symmetry breaking operators for principal series representations of the pair $(S{O}_0(4,1),S{O}_0(3,1))$ for special parameters

We construct and give a complete classification of all the differential symmetry breaking operators ${\mathbb{D}}_{\lambda ,\nu }^{N,m}:C^{\mathrm{\infty }}(S^3,{\mathcal{V}}_{\lambda }^{2N+1})\to C^{\mathrm{\infty }}(S^2,{\mathcal{L}}_{m,\nu })$, between the spaces of smooth sections of a vector bundle of rank $2N+1$ over the $3$-sphere ${\mathcal{V}}_{\lambda }^{2N+1}\to S^3$, and a line bundle over the $2$-sphere ${\mathcal{L}}_{m,\nu }\to S^2$ in the special case $\mathrm{\mid }m\mathrm{\mid }=N$.