The second Hamiltonian structure for a special case of the Lotka–Volterra equations

A special case of the Lotka–Volterra equations is considered for which it is possible to find the second Hamiltonian structure that is complementary to the known one. The form of the new Hamiltonian makes it possible to solve the equations by quadratures, which is the main feature of the case under examination. As a consequence, the period can also be represented by quadratures. In terms of the new variables, the equations of motion admit a mechanical analogy with the oscillations of a mass on a nonlinear spring.