A Lie algebra structure on variation vector fields along curves in   $2$-dimensional space forms
  A Lie algebra structure on variation vector fields along an immersed curve in a $2$-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure for plane curve motions. The Hamiltonian form and the integrability of the planar filament equation is finally discussed from this point of view. 
Jos\'e del Amor, \'Angel Gim\'enez and Pascual Lucas
math.DG