The aim of this paper is to deal with the k-Hessian counterpart of the Laplace equation involving a nonlinearity studied by Matukuma. Namely, our model is the problem(1) {Sk(D(2)u) = lambda vertical bar x vertical bar (mu-2)/(1+vertical bar x vertical bar(2))(mu/2) (1 - u)(q) in B,u < 0 in B,u = 0 on partial derivative B,where B denotes the unit ball in R-n, n > 2k (k is an element of N), lambda > 0 is an additional parameter, q > k and mu >= 2. In this setting, through a transformation recently introduced by two of the authors that reduces problem (1) to a non-autonomous two-dimensional generalized Lotka-Volterra system, we prove the existence and multiplicity of solutions for the above problem combining dynamical-systems tools, the intersection number between a regular and a singular solution and the super and subsolution method.