  We consider Dunkl version of Laplace-Runge-Lenz vector associated with a
finite Coxeter group $W$ acting geometrically in $\mathbb R^N$ with
multiplicity function $g$. This vector generalizes the usual Laplace-Runge-Lenz
vector and its components commute with Dunkl-Coulomb Hamiltonian given as Dunkl
Laplacian with additional Coulomb potential $\gamma/r$. We study resulting
symmetry algebra $R_{g, \gamma}(W)$ and show that it has
Poincar\'e-Birkhoff-Witt property. In the absence of Coulomb potential this
symmetry algebra $R_{g,0}(W)$ is a subalgebra of the rational Cherednik algebra
$H_g(W)$. We show that a central quotient of the algebra $R_{g, \gamma}(W)$ is
a quadratic algebras isomorphic to a central quotient of the corresponding
Dunkl angular momenta algebra $H_g^{so(N+1)}(W)$. This gives interpretation of
the algebra $H_g^{so(N+1)}(W)$ as the hidden symmetry algebra of Dunkl-Coulomb
problem in $\mathbb R^N$. By specialising $R_{g, \gamma}(W)$ to $g=0$ we
recover a quotient of the universal enveloping algebra $U(so(N+1))$ as the
hidden symmetry algebra of Coulomb problem in $\mathbb R^N$. We also apply
Dunkl Laplace-Runge-Lenz vector to establish maximal superintegrability of
generalised Calogero-Moser systems.
