2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/159603We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamiltonian operators. The Lagrangian formulation, via Noether's theorem, provides a method to make the relation between symmetries and conserved quantities more precise. We have exploited this fact to study the variational symmetries of the dispersionless KdV equation.Published in SIGMA (Symmetry, Integrability and Geometry: Methods and pplications) at http://www.emis.de/journals/SIGMA/Exactly Solvable and Integrable Systemstext