  In an uncoupled linear lattice system, the Kapchinskij-Vladimirskij (KV)
distribution, formulated on the basis of the single-particle Courant-Snyder
(CS) invariants, has served as a fundamental theoretical basis for the analyses
of the equilibrium, stability, and transport properties of high-intensity beams
for the past several decades. Recent applications of high-intensity beams,
however, require beam phase-space manipulations by intentionally introducing
strong coupling. In this Letter, we report the full generalization of the KV
model by including all of the linear (both external and space-charge) coupling
forces, beam energy variations, and arbitrary emittance partition, which all
form essential elements for phase-space manipulations. The new generalized KV
model yields spatially uniform density profiles and corresponding linear
self-field forces as desired. The corresponding matrix envelope equations and
beam matrix for the generalized KV model provide important new theoretical
tools for the detailed design and analysis of high-intensity beam
manipulations, for which previous theoretical models are not easily applicable.
