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

The Watanabe-Strogatz transform and constant of motion functionals for kinetic vector models