  This paper is devoted to a systematic study of the geometry of nondegenerate
$\bbR^n$-actions on $n$-manifolds. The motivations for this study come from
both dynamics, where these actions form a special class of integrable dynamical
systems and the understanding of their nature is important for the study of
other Hamiltonian and non-Hamiltonian integrable systems, and geometry, where
these actions are related to a lot of other geometric objects, including
reflection groups, singular affine structures, toric and quasi-toric manifolds,
monodromy phenomena, topological invariants, etc. We construct a geometric
theory of these actions, and obtain a series of results, including: local and
semi-local normal forms, automorphism and twisting groups, the reflection
principle, the toric degree, the monodromy, complete fans associated to
hyperbolic domains, quotient spaces, elbolic actions and toric manifolds,
existence and classification theorems.
