The paper proposes a double excited synchronous machine as integrated starter-alternator for future automobiles applications. Particularly, the biaxial excitation synchronous machine (BESM) is proposed for its high efficiency and wide constant power speed range. As the integrated starter-alternator system combines both starter and generator functions in a single electric machine, an alternative vector control strategy for BESM is developed and detailed to allow operation in the two modes at unity power factor operation by setting d-axis current to be zero and q-axis current to be constant and equal to its value that cancel the q-axis flux. In such conditions, the BESM torque can be controlled by the dc-excitation current and the required dc-bus voltage can be regulated within the 42V PowerNet specifications. To overcome the slowness dc-excitation current response, which is due to higher dc-field excitation time constant, and to increase the torque response quickness, the control of the magnetising current is proposed instead of the d-axis current. Implementation and simulation results validate the proposed scheme and provide a practical solution for an integrated starter-alternator.