Systems of Newton equations of the formq = − 1 2 A −1 (q)∇k with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A=I, the identity matrix) and they admit a curious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by embedding into five-dimensional integrable systems. They are characterized by a linear, second-order PDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of non-confocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamical systems of the Hénon-Heiles type.

A systematic method of constructing finite-dimensional integrable systems starting from a bi-Hamiltonian hierarchy of soliton equations is introduced. The existence of two Hamiltonian structures of the hierarchy leads to a bi-Hamiltonian formulation of the resulting finite-dimensional systems. The case of coupled KdV hierarchies is studied in detail. A surprising connection with separable Jacobi potentials is uncovered and described.

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Note on the Poisson structure of the damped oscillatorTwo families of nonstandard two-dimensional Poisson structures for systems of Newton equations are studied. They are closely related either with separable systems or with the so-called quasi-Lagrangian systems. A theorem characterizing the general form of bi-Hamiltonian formulation for separable systems in two and in n dimensions is formulated and proved.

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