  The Hamilton-Jacobi formalism for a geodetic brane-like universe described by
the Regge-Teitelboim model is developed. We focus on the description of the
complete set of Hamiltonians that ensure the integrability of the model in
addition to obtaining the Hamilton principal function $S$. In order to do this,
we avoid the second-order in derivative nature of the model by appropriately
defining a set of auxiliary variables that yields a first-order Lagrangian.
Being a linear in acceleration theory, this scheme unavoidably needs an
adequate redefinition of the so-called Generalized Poisson Bracket in order to
achieve the right evolution in the reduced phase space. Further this
Hamilton-Jacobi framework also enables us to explore the quantum behaviour
under a semi-classical approximation of the model. A comparison with the
Ostrogradski-Hamilton method for constrained systems is also provided in
detail.
