  By introducing a basis for a novel realization of the SU(4) Lie algebra, we
exactly solve a spin-orbital chain with one-ion $L$-$S$ coupling (OILSC) via
the Bethe ansatz (BA) approach. In the context of different Land\'e $g$ factors
of the spin and orbital sectors, the OILSC results in rich and novel quantum
phase transitions. Some accurate analytical expressions for the critical fields
are obtained. Both spin ordering and orbital ordering are found in a gapped
singlet phase. The system exhibits many interesting phenomena such as
nonvanishing magnetization of the singlet phase, multi-entrance of the singlet
in the ground state when the field varies, unsymmetric magnetization in
two-component phase and magnetization crossover for different OILSC.
