Some constructions of integrable dynamical systems

New constructions of integrable dynamical systems are found that admit representation as Lax matrix equations. A countable set of integrable systems is constructed which in the continuous limit turn into the Korteweg–de Vries equation. For an arbitrary space $\mathscr M$ with finite measure $\mu$ and measure-preserving mapping $\alpha\colon\mathscr M\to\mathscr M$ differential equations are constructed on the space of measurable functions on $\mathscr M$. Here differentiation is with respect to time $t$ and the equations have a countable set of first integrals. Constructions are also given for first integrals of dynamical systems preserving certain differential forms, and new constructions of matrix differential equations having large families of first integrals.
Bibliography: 18 titles.