Deformations of vector fields and Hamiltonian vector fields on the plane

For the Lie algebras L1 (H(2)) and L1 ( W(2)), we study their infinitesimal deformations and the corresponding global ones. We show that, as in the case of L1(W(l)), each integrable infinitesimal deformation of L1(H(2)) and L1 (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 L1 (H(2)) arise as restrictions of deformations of Li( W(2)).