<p>This paper is about the existence of periodic orbits near an equilibrium point of a two-degree-of-freedom Hamiltonian system. The equilibrium is supposed to be a nondegenerate minimum of the Hamiltonian. Every sphere-like component of the energy surface sufficiently close to the equilibrium contains at least two periodic orbits forming a Hopf link (Weinstein in Inv Math 20:47–57, 1973). A theorem by Hofer, Wysocki and Zehnder (Ann Math 148:197–289, 1998) implies that there are either precisely two or infinitely many periodic orbits on such a component of the energy surface. This multiplicity result follows from the existence of a disk-like global surface of section. If a certain non-resonance condition on the rotation numbers of the orbits of the Hopf link is satisfied, then by Hryniewicz, Momin and Salomão (Inv Math 199:333–422, 2015), infinitely many periodic orbits follow. This paper aims to present explicit conditions on the Birkhoff-Gustavson normal forms of the Hamiltonian function at the equilibrium point that ensure the existence of infinitely many periodic orbits on the energy surface by checking the non-resonance condition as in (Hryniewicz, Momin and Salomão in Inv Math 199:333–422, 2015) and not making use of any global surface of section. The main results apply to the Spatial Isosceles Three-Body Problem, Hill’s Lunar Problem, and the Hénon-Heiles System, where non-resonant Hopf links are proved to exist.</p>