  In this paper we develop the vectorial Ribaucour transformation for Euclidean
submanifolds. We prove a general decomposition theorem showing that under
{appropriate} conditions the composition of two or more vectorial Ribaucour
transformations is again a vectorial Ribaucour transformation. An immediate
consequence of this result is the classical permutability of Ribaucour
transformations. Our main application is an explicit local construction of all
Euclidean submanifolds with flat normal bundle. Actually, this is a particular
case of a more general result. Namely, we obtain a local explicit construction
of all Euclidean submanifolds carrying a parallel flat normal subbundle, in
particular of all those that carry a parallel normal vector field. Finally, we
describe all submanifolds carrying a Dupin principal curvature normal vector
field with integrable conullity, a concept that has proven to be crucial in the
study of reducibility of Dupin submanifolds.
