The purpose of Very Special Relativity is to show that the ISIM(2) subgroup of the Poincaré group is sufficient to describe the spacetime symmetries of the so far observed physical phenomena. A deformation of such group, called DISIMb(2), was later introduced. In the present work, we present a novel non-commutative spacetime structure, underlying the DISIMb(2), that allows us to construct explicitly the generators of the group. Exploiting the Darboux map technique, we then construct a point particle Lagrangian that lives in the non-commutative phase space proposed by us.
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