A study of the consistency between noncommutative quantum mechanics and   Galilean isotropy
  A demonstration is given that the simplest model of quantum mechanics formulated on a plane non-commutative geometry endowed with a Galilean symmetry group in which the position and linear momentum-variable commutators are first order in the dynamical variables (and thus constitute a true Lie algebra) is incompatible with the hypothesis of spacial isotropy. 
Jose Ignacio Usera