  We investigate one-dimensional charge conserving, spin-singlet (SSS) and
spin-triplet (STS) superconductors in the presence of boundary fields. In
systems with Open Boundary Conditions (OBC) it has been demonstrated that STS
display a four-fold topological degeneracy, protected by the $\mathbb{Z}_2$
symmetry which reverses the spins of all fermions, whereas SSS are
topologically trivial. In this work we show that it is not only the type of the
bulk superconducting instability that determines the eventual topological
nature of a phase, but rather the interplay between bulk and boundary
properties. In particular we show by means of the Bethe Ansatz technique that
SSS may as well be in a $\mathbb{Z}_2$-protected topological phase provided
suitable "twisted" open boundary conditions ${\widehat{OBC}}$ are imposed. More
generally, we find that depending on the boundary fields, a given
superconductor, either SSS or STS, may exhibits several types of phases such as
topological, mid-gap and trivial phases; each phase being characterized by a
boundary fixed point which which we determine. Of particular interest are the
mid-gap phases which are stabilized close to the topological fixed point. They
include both fractionalized phases where spin-$\frac{1}{4}$ bound-states are
localized at the two edges of the system and un-fractionalized phases where a
spin-$\frac{1}{2}$ bound-state is localized at either the left or the right
edge.
