  Self-similar solutions of the so called Airy equations, equivalent to the
dispersionless nonlinear Schr\"odinger equation written in Madelung
coordinates, are found and studied from the point of view of complete
integrability and of their role in the recurrence relation from a
bi-Hamiltonian structure for the equations. This class of solutions reduces the
PDEs to a finite ODE system which admits several conserved quantities, which
allow to construct explicit solutions by quadratures and provide the
bi-Hamiltonian formulation for the reduced ODEs.
