  We show that in the presence of magnetic field, two superconducting phases
with the center-of-mass momentum of Cooper pair parallel to the magnetic field
are induced in spin-orbit-coupled superconductor Li$_2$Pd$_3$B. Specifically,
at small magnetic field, the center-of-mass momentum is induced due to the
energy-spectrum distortion and no unpairing region with vanishing singlet
correlation appears. We refer to this superconducting state as the drift-BCS
state. By further increasing the magnetic field, the superconducting state
falls into the Fulde-Ferrell-Larkin-Ovchinnikov state with the emergence of the
unpairing regions. The observed abrupt enhancement of the center-of-mass
momenta and suppression on the order parameters during the crossover indicate
the first-order phase transition. Enhanced Pauli limit and hence enlarged
magnetic-field regime of the Fulde-Ferrell-Larkin-Ovchinnikov state, due to the
spin-flip terms of the spin-orbit coupling, are revealed. We also address the
triplet correlations induced by the spin-orbit coupling, and show that the
Cooper-pair spin polarizations, generated by the magnetic field and
center-of-mass momentum with the triplet correlations, exhibit totally
different magnetic-field dependences between the drift-BCS and
Fulde-Ferrell-Larkin-Ovchinnikov states.
