  We study a one-dimensional Kitaev model with uniform phase gradient pairing
term. We show that the gradient constant dramatically affects the phase
diagram, which consists of topologically trivial and nontrivial phases,
associated with Majorana edge modes. Based on the exact solution, a
Bardeen-Cooper-Schrieffer (BCS)-pair order parameter is introduced to
characterize the phase diagram by its value and nonanalytic behavior at phase
boundaries. We find that this order parameter obtains its maxima at the triple
critical points, at which the pairing phase gradient suppresses the
single-particle scattering process due to the coherent destructive
interference. In particular, we show that the ground state at such a point
possesses exact off-diagonal long-range order (ODLRO), in the thermodynamic
limit. Our result provides an example of a gapless $p$-wave superconducting
ground state possessing ODLRO.
