It is shown how an integrable mechanical system provides all the localized static solutions of a deformation of the linear O(N)-sigma model in two spacetime dimensions. The proof is based on the Hamilton-Jacobi separability of the mechanical analogue system that follows when time-independent field configurations are being considered. In particular, we describe the properties of the different kinds of kinks in such a way that a hierarchical structure of solitary wave manifolds emerges for distinct N.