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summary:We introduce the symplectic twistor operator $T_s$ in symplectic spin geometry of real dimension two, as a symplectic analogue of the Dolbeault operator in complex spin geometry of complex dimension 1. Based on the techniques of the metaplectic Howe duality and algebraic Weyl algebra, we compute the space of its solutions on ${\mathbb{R}}^2$

On modules of bounded multiplicities for the symplectic algebras