Polynomial integrals of dynamical systems with one-and-a-half degrees of freedom

On integrability of a heavy rigid body sinking in an ideal fluid

“…We have demonstrated a specific property of symmetric rigid body dynamics in an ideal fluid, namely that the general solution always branches in the complex time plane. This extends the result of Kozlov on the nonexistence of polynomial first integral for Chaplygin's equation [14], [17].…”

“…We show that in fact there are solutions with an infinitely-sheeted Riemannian surface. This extends and complements Kozlov's results [14] on non-existence of a polynomial first integral for Chaplygin's equation.…”

“…However the work [10] is not applicable to our case, as equations (3)-(5) are Hamiltonian, which implies resonances between the pairs of eigenvalues (see [16]). Theorem 3.1 extends the results of Kozlov on nonexistence of first integral polynomial in velocities for Chaplygin's equation [14], which is in fact the system (8) after the transformation…”

Algebro-Geometric Approach in the Theory of Integrable Hydrodynamic Type Systems

“…Nevertheless, the first step is a description of N series of conservation laws (see (17)). They can be obtained by expansion in the Bürmann-Lagrange series (see, for instance, [22]) at the vicinity of each singular point.…”

In search of periodic solutions for a reduction of the Benney chain

“…It was observed in [14] and later in [4], [5], [6] that the question of existence of smooth periodic solution for (1) is ultimately related to the search of polynomial integrals for a Hamiltonian system as we now turn to explain.…”