Near-Kerr black hole initial datasets are constructed by applying either the
parabolic-hyperbolic or the algebraic-hyperbolic form of the constraints. In
both cases, strongly and weakly asymptotically flat initial datasets with
desirable falloff rates are produced by controlling only the monopole part of
one of the freely specifiable variables. The viability of the applied method is
verified by numerically integrating the evolutionary forms of the constraint
equations in the case of various near-Kerr configurations.Comment: 43 pages, 6 figure