  In this article we introduce the notion of a Ribaucour partial tube and use
it to derive several applications. These are based on a characterization of
Ribaucour partial tubes as the immersions of a product of two manifolds into a
space form such that the distributions given by the tangent spaces of the
factors are orthogonal to each other with respect to the induced metric, are
invariant under all shape operators, and one of them is spherical. Our first
application is a classification of all hypersurfaces with dimension at least
three of a space form that carry a spherical foliation of codimension one,
extending previous results by Dajczer, Rovenski and the second author for the
totally geodesic case. We proceed to prove a general decomposition theorem for
immersions of product manifolds, which extends several related results. Other
main applications concern the class of hypersurfaces of $\mathbb{R}^{n+1}$ that
are of Enneper type, that is, hypersurfaces that carry a family of lines of
curvature, correspondent to a simple principal curvature, whose orthogonal
$(n-1)$-dimensional distribution is integrable and whose leaves are contained
in hyperspheres or affine hyperplanes of $\mathbb{R}^{n+1}$. We show how
Ribaucour partial tubes in the sphere can be used to describe all
$n$-dimensional hypersurfaces of Enneper type for which the leaves of the
$(n-1)$-dimensional distribution are contained in affine hyperplanes of
$\mathbb{R}^{n+1}$, and then show how a general hypersurface of Enneper type
can be constructed in terms of a hypersurface in the latter class. We give an
explicit description of some special hypersurfaces of Enneper type, among which
are natural generalizations of the so called Joachimsthal surfaces.
