  We construct and give a complete classification of all the differential
symmetry breaking operators D_{{\lambda},{\nu}}^m : C^\infty(S^3,
V^3_{\lambda}) \rightarrow C^\infty(S^2,L_{m,{\nu}}), between the spaces of
smooth sections of a vector bundle of rank 3 over the 3-sphere V^3_{\lambda}
\rightarrow S^3, and a line bundle over the 2-sphere L_{m,{\nu}} \rightarrow
S^2. In particular, we give necessary and sufficient conditions on the tuple of
parameters ({\lambda}, {\nu}, m) for which these operators exist.
