A class of submanifolds in $V_c^n$ with parallel $p$-dimensional subbundle of the normal bundle

The local structure of a submanifold $V^m$ is studied for which the focal surface $F$ of a subbundle $N^p$ has $s$ distinct components with multiplicities $p_1\dots,p_n$ ($p_1+\dots+p_s=m$), and the focal surface $\Phi$ of the subbundle $N^{n-m-p}$ has no multiple components.