  In this paper we study the exact solution of a one-dimensional model of
spin-$\frac{1}{2}$ electrons composed by a nearest-neighbor triplet pairing
term and the on-site Hubbard interaction. We argue that this model admits a
Bethe ansatz solution through a mapping to a Hubbard chain with imaginary
kinetic hopping terms. The Bethe equations are similar to that found by Lieb
and Wu \cite{LW} but with additional twist phases which are dependent on the
ring size. We have studied the spectrum of the model with repulsive interaction
by exact diagonalization and through the Bethe equations for large lattice
sizes. One feature of the model is that it is possible to define the charge gap
for even and odd lattice sites and both converge to the same value in the
infinite size limit. We analyze the finite-size corrections to the low-lying
spin excitations and argue that they are equivalent to that of the
spin-$\frac{1}{2}$ isotropic Heisenberg model with a boundary twist depending
on the lattice parity. We present the classical statistical mechanics model
whose transfer matrix commutes with the model Hamiltonian. To this end we have
used the construction employed by Shastry \cite{SHA1,SHA2} for the Hubbard
model. In our case, however, the building block is a free-fermion eight-vertex
model with a particular null weight.
