  Following Dirac's brane variation prescription, the brane must not be
deformed during the variation process, or else the linearity of the variation
may be lost. Alternatively, the variation of the brane is done, in a special
Dirac frame, by varying the bulk coordinate system itself. Imposing appropriate
Dirac style boundary conditions on the constrained 'sandwiched' gravitational
action, we show how Israel junction conditions get relaxed, but remarkably, all
solutions of the original Israel equations are still respected. The Israel
junction conditions are traded, in the $Z_2$-symmetric case, for a generalized
Regge-Teitelboim type equation (plus a local conservation law), and in the
generic $Z_2$-asymmetric case, for a pair of coupled Regge-Teitelboim
equations. The Randall-Sundrum model and its derivatives, such as the
Dvali-Gabadadze-Porrati and the Collins-Holdom models, get generalized
accordingly. Furthermore, Randall-Sundrum and Regge-Teitelboim brane theories
appear now to be two different faces of the one and the same unified brane
theory. Within the framework of unified brane cosmology, we examine the dark
matter/energy interpretation of the effective energy/momentum deviations from
General Relativity.
