Bäcklund transformations for new fourth Painlevé hierarchies

We consider a system of equations defined using the Hamiltonian operator of the Boussinesq hierarchy, as well as two successive modifications thereof. We are able to reduce the order of these three systems and give Bäcklund transformations between the integrated equations. We also give auto-Bäcklund transformations for the two modified systems.Particular cases of two of the three equations considered correspond to generalized fourth Painlevé hierarchies and are new; these are particular cases of the two modified systems. Thus we obtain auto-Bäcklund transformations for these new fourth Painlevé hierarchies, as well as Bäcklund transformations between our hierarchies. Our results on reduction of order are also applicable in this special case, and include as a particular example a reduction of order for the scaling similarity reduction of the Boussinesq equation, a result which, remarkably, seems not to have been given previously.