Homo- and heteroclinic orbits, hyperbolic subsets in a one-parameter unfolding of a Hamiltonian system with heteroclinic contour with two saddle-foci

We study a $1$-parametric family of the Hamiltonian systems with $2$ hyperbolic fixed points and analyze the structure and bifurcations of homoclinic and heteroclinic trajectories under the variation of the parameter and energy values.