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$N$ "exotic" [alias non-commutative] particles with masses $m_a$, charges$e_a$ and non-commutative parameters $\theta_a$, moving in a uniform magneticfield $B$, separate into center-of-mass and internal motions if Kohn'scondition $e_a/m_a=\const$ is supplemented with $e_a\theta_a=\const.$ Then thecenter-of-mass behaves as a single exotic particle carrying the total mass andcharge of the system, $M$ and $e$, and a suitably defined non-commutativeparameter $\Theta$. For vanishing electric field off the critical case $e\ThetaB\neq1$, the particles perform the usual cyclotronic motion with modified butequal frequency. The system is symmetric under suitable time-dependenttranslations which span a (4+2)- parameter centrally extended subgroup of the"exotic" [i.e., two-parameter centrally extended] Newton-Hooke group. In thecritical case $B=B_c=(e\Theta)^{-1}$ the system is frozen into a static"crystal" configuration. Adding a constant electric field, all particlesperform, collectively, a cyclotronic motion combined with a drift perpendicularto the electric field when $e\Theta B\neq1$. For $B=B_c$ the cyclotronic motionis eliminated and all particles move, collectively, following the Hall law. Ourtime-dependent symmetries are reduced to the (2+1)-parameter Heisenberg groupof centrally-extended translations.

Kohn condition and exotic Newton-Hooke symmetry in the non-commutative Landau problem