A generalization of the classical Leray-Schauder fixed-point theorem based on the infinite-dimensional Borsuk-Ulam-type antipode construction is proposed. A new nonstandard proof of the classical LeraySchauder fixed-point theorem and a study of the solution manifold of a nonlinear Hamilton-Jacobi-type equation are presented.

A Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems is obtained via some special Båcklund transformation. The connection of this hierarchy with Lax-integrable twometrizable systems is studied.

A Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems is obtained via some special Båcklund transformation. The connection of this hierarchy with Lax-integrable twometrizable systems is studied.

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