  We discuss the Hamilton-Jacobi formalism for brane gravity described by the
Regge-Teitelboim model, in higher co-dimension. Being originally a second-order
in derivatives singular theory, we analyzed its constraint structure by
identifying the complete set of Hamilton-Jacobi equations, under the
Carath\'eodory's equivalent Lagrangians method, which goes hand by hand with
the study of the integrability for this type of gravity. Besides, we calculate
the characteristic equations including the one that satisfy the Hamilton
principal function $S$. We find the presence of involutive and non-involutive
constraints so that by properly defining a generalized bracket, the
non-involutive constraints that originally arise in our framework, are removed
while the set of parameters related to the time evolution and the gauge
transformations, are identified. A detailed comparison with a recent
Ostrogradsky-Hamilton approach for constrained systems, developed for this
brane gravity, is also made. Some remarks on the gauge symmetries behind this
theory are commented upon.
