Core Concepts
The article explores the properties of extended integrable field theories and multiparticle systems formulated in terms of the root systems of infinitedimensional KacMoody algebras, including hyperbolic and Lorentzian types.
Abstract
The article discusses the mathematical framework and physical properties of extended integrable models based on infinite Weyl groups:

Nextended Lorentzian KacMoody algebras:
 The author introduces a Lorentzian inner product and an extended root system to define Lorentzian KacMoody algebras.
 The Cartan matrix, weight lattice, Weyl reflections, and Coxeter elements are constructed for these extended algebras.
 The existence of SO(2,1) principal subalgebras is shown to be a necessary condition for integrability.

L̊gextended Lorentzian Toda Field Theories:
 The author extends the standard Toda field theories by consecutively adding specific roots, obtaining massless and massive models.
 The mass spectrum and integrability properties of the L̊gextended Toda theories are investigated, showing that only the affine case is integrable.

Extended Calogero Models:
 The Calogero models are generalized to include the entire root space of the extended KacMoody algebras.
 For the example of the doubly extended A2 affine algebra, the author derives closedform expressions for the potential by identifying the representatives of the Coxeter orbits.
 The potential is shown to be invariant under the infinitedimensional affine Weyl group.
The article highlights many open mathematical and physical questions regarding the properties of these extended integrable models.