  We present a kinetic version of the Watanabe-Strogatz(WS) transform for
vector models in this paper. From the generalized WS-transform, we obtain the
cross-ratio type constant of motion functionals for kinetic vector models under
suitable conditions. We present the sufficient and necessary conditions for the
existence of the suggested constant of motion functional. As an application of
the constant of motion functional, we provide the instability of bipolar states
of the kinetic swarm sphere model. We also provide the WS-transform and
constant of motion functional for non-identical kinetic vector models.
