  In this paper we present results obtained from the unification of $SU(2)$
coherent states with $\mathbb{C}P^N$ sigma models defined on the Riemann sphere
having finite actions. The set of coherent states generated by a vector
belonging to a carrier space of an irreducible representation of the group
gives rise to a map from the sphere into the set of rank-1 Hermitian projectors
in that space. The map can be identified with a particular solution of the
$\mathbb{C}P^N$ sigma model, where $N+1$ is equal to the dimension of the
representation space. In particular a choice of the generating vector as the
highest weight vector of the representation gives rise to the map known as a
Veronese immersion. Using a description of the matrix elements of these
representations in terms of Jacobi polynomials, we obtain an explicit
parametrization of the solutions of the $\mathbb{C}P^N$ models, which has not
been previously found. We relate the analytical properties of the solutions,
which are known to belong to separate classes --- holomorphic, anti-holomorphic
and various types of mixed ones --- to the weight corresponding to the chosen
weight vector. Some examples of the described constructions are elaborated in
detail in this paper.
