Vibrations of plates, shells and plate}shell systems coupled with sloshing, quiescent and inviscid #uid have been advantageously studied by inserting the sloshing condition into the eigenvalue problem. Here a formulation of this particular eigenvalue problem for symmetric matrices is obtained. In fact, in the previous studies, this technique has given eigenvalue problems for non-symmetric matrices for which the problem of the existence of complex eigenvalues arises. The present analysis deals with compressible and incompressible #uids and the discretization of the system is obtained by using the Rayleigh}Ritz method. The Rayleigh quotient of the system is manipulated to obtain expressions suitable for symmetric formulations of the eigenvalue problem. In particular, the Rayleigh quotient is transformed into a simpler expression where the potential energies of the compressible #uid and free surface waves do not appear. The method is applied to a vertical, simply supported, circular cylindrical shell partially "lled by an incompressible sloshing liquid. A case with large interaction between sloshing and bulging modes is considered and interesting phenomena are observed.