<p>The Hamilton equations of a system with two degrees of freedom are considered, where the real Hamilton function depends on four real independent variables. At a fixed point (equilibrium point) the Hamilton function <i>H</i> is analytic and can be expanded into a power series in independent variables. We consider the case when the characteristic polynomial of the Hamiltonian equations with quadratic part of Hamiltonian has two pairs of purely imaginary roots in resonance 2:1. Then there exists a symplectic linear transformation that reduces the Hamiltonian to a complex normal form. The expansion of the normal form is represented by a sum of homogeneous polynomials of independent variables. We consider its truncated part consisting of quadratic and cubic terms. The truncated normal form has two integrals: preservation of the quadratic and cubic parts of the normal form. By Liouville’s theorem it is integrable in quadratures. This paper presents exact analytical solutions to the Hamiltonian system defined by the truncated normal form. Complex variables are expressed in terms of Jacobi elliptic functions, and under some particular initial conditions—in terms of elementary functions. The obtained results are verified on two physical examples: a swinging spring and a motion around a triangular libration point.</p>