  We introduce a new class of generalized isotropic Lipkin-Meshkov-Glick models
with su$(m+1)$ spin and long-range non-constant interactions, whose
non-degenerate ground state is a Dicke state of su$(m+1)$ type. We evaluate in
closed form the reduced density matrix of a block of $L$ spins when the whole
system is in its ground state, and study the corresponding von Neumann and
R\'enyi entanglement entropies in the thermodynamic limit. We show that both of
these entropies scale as $a\log L$ when $L$ tends to infinity, where the
coefficient $a$ is equal to $(m-k)/2$ in the ground state phase with $k$
vanishing su$(m+1)$ magnon densities. In particular, our results show that none
of these generalized Lipkin-Meshkov-Glick models are critical, since when
$L\to\infty$ their R\'enyi entropy $R_q$ becomes independent of the parameter
$q$. We have also computed the Tsallis entanglement entropy of the ground state
of these generalized su$(m+1)$ Lipkin-Meshkov-Glick models, finding that it can
be made extensive by an appropriate choice of its parameter only when
$m-k\ge3$. Finally, in the su$(3)$ case we construct in detail the phase
diagram of the ground state in parameter space, showing that it is determined
in a simple way by the weights of the fundamental representation of su$(3)$.
This is also true in the su$(m+1)$ case; for instance, we prove that the region
for which all the magnon densities are non-vanishing is an $(m+1)$-simplex in
$\mathbf R^m$ whose vertices are the weights of the fundamental representation
of su$(m+1)$.
