  We present a family of many-body models which are exactly solvable
analytically. The models are an extended n-body interaction
Lipkin-Meshkov-Glick model which considers spin-flip terms which are associated
with the interaction of an external classical field which coherently
manipulates the state of the system in order to, for example, process quantum
information. The models also describe a two-mode Bose-Einstein condensate with
a Josephson-type interaction which includes n-particle elastic and inelastic
collisions. One of the models corresponds to the canonical two-mode
Bose-Einstein Hamitonian plus a term which we argue must be considered in the
description of the two-mode condensate. Intriguingly, this extra term allows
for an exact and analytical solution of the two-particle collision two-mode BEC
problem. Our results open up an arena to study many-body system properties
analytically.
