  We prove that for a large class of generalized Randall-Sundrum II type models
the characterization of brane-gravity sector by the effective Einstein
equation, Codazzi equation and the twice-contracted Gauss equation is
equivalent with the bulk Einstein equation. We give the complete set of
equations in the generic case of non-$Z_{2}$-symmetric bulk and arbitrary
energy-momentum tensors both on the brane and in the bulk. Among these, the
effective Einstein equation contains a varying cosmological constant and two
new source terms. The first of these represents the deviation from $Z_{2}$
symmetry, while the second arises from the bulk energy-momentum tensor. We
apply the formalism for the case of perfect fluid on a Friedmann brane embedded
in a generic bulk. The generalized Friedmann and Raychaudhuri equations are
given in a form independent of both the embedding and the bulk matter. They
contain two new functions obeying a first order differential system, both
depending on the bulk matter and the embedding. Then we focus on Friedmann
branes separating two non-identical (inner or outer) regions of
Reissner-Nordstr\"om-Anti de Sitter bulk space-times, generalizing previous
non-$Z_{2}$-symmetric treatments. Finally the analysis is repeated for the
Vaidya-Anti de Sitter bulk space-time, allowing for both ingoing and outgoing
radiation in each region.
