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Deformations of vector fields and Hamiltonian vector fields on the plane

For the Lie algebras L 1 (H(2)) and L 1 (W(2)), we study their infinitesimal deformations and the corresponding global ones. We show that, as in the case of L 1 (W(1)), each integrable infinitesimal deformation of L 1 (H(2)) and L 1 (W(2)) can be represented by a 2-cocycle that defines a global deformation by means of a trivial extension. We also illustrate that all deformations of L 1 (H(2)) arise as restrictions of deformations of L 1 (W(2)).