A Family of Second Lie Algebra Structures for Symmetries of Dispersionless Boussinesq System

for (1), see [1], transfer the standard bracket [ , ] in sym E to the Lie algebra structures on their domain. We prove that the three new brackets are compatible. The Noether operator A0 is invertible on an open dense subset of E . This yields two recursion operators Ri = Âi ◦ A −1 0 : sym E → sym E . The images of Ri are again closed w.r.t. the commutation, and this property is retained by their arbitrary linear combinations. Using the ‘chain rule’ formula (11) for the bi-differential brackets on domains of the operators Âi and Ri, we calculate the second Lie algebra structures [ , ]Ri on sym E . All notions and constructions are standard [3, 4]. We stress that the concept of linear compatible differential operators with involutive images, which we develop here, can be applied to the study of other integrable systems with or without dispersion (e.g., see [5]).