ABSTRACT
The Toda systems permanently attract attention of physicists and mathematicians due to their integrability and deep links to a number of problems in the theory of differential equations, differential and algebraic geometry, and relevance to many problems of modern theoretical and mathematical physics. This chapter reviews the construction of the Toda-type equations in low and high dimensional spaces and discusses the integration procedure for these equations. It presents a differential geometry formulation of the main principle of the group-algebraic approach to the integration of the systems under consideration, the so-called grading condition. The chapter examines the proof of the generalised Pliicker formulae, based on the use of Toda equations.
