The paper investigates a growth model with a production function of constant elasticity of substitution, which generalizes such functions as Cobb-Douglas or Leontief, when one of its parameter tends to zero or infinity respectively. The investment indicators of the model are considered as control parameters that are chosen in order to maximize the utility functional. An optimal control problem with an infinite planning horizon is formulated. Using the Pontryagin maximum principle, the paper studies a Hamiltonian function and a Hamiltonian system and provides its qualitative analysis. Next, the existence and uniqueness of a steady state are proven, and an algorithm for its search is given by solving a nonlinear equation of one special variable. The following section performs the stabilization of the Hamiltonian system in the vicinity of the steady state using a controller that can be constructed due to saddle character of the steady state. Finally, a numerical example is given that illustrates obtained analytical results.