  The issues related to the integrability of quantum Calogero-Moser models
based on any root systems are addressed. For the models with degenerate
potentials, i.e. the rational with/without the harmonic confining force, the
hyperbolic and the trigonometric, we demonstrate the following for all the root
systems: (i) Construction of a complete set of quantum conserved quantities in
terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal
R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the
Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of
the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack
polynomials are defined for all root systems as unique eigenfunctions of the
Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v)
Algebraic construction of all excited states in terms of creation operators.
These are mainly generalisations of the results known for the models based on
the (A) series, i.e. (su(N)) type, root systems.
