Nonlinear integrable systems represent a very important and popular branch of theoretical and mathematical physics, and most of the famous universities and colleges currently include this subject in their educational programmes for students and post-graduate students of physical, mathematical, and even technical specialities. Over the last decade in particular, investigations related to studies of nonlinear phenomena have been in the foreground in an overwhelming majority of areas of modern theoretical and mathematical physics, especially in elementary particle, solid state and plasma physics, nonlinear optics, physics of the Earth, etc. The principal physical properties resulting from the nonlinear nature of the phenomena itself are not in general reproduced here by perturbative methods. This fact leads to the necessity to construct the exact solutions of the corresponding nonlinear differential equations describing the dynamical systems under consideration.

To the present time, physics has placed at our disposal a wide range of nonlinear equations arising repeatedly in its various branches. The methods of their explicit integration began to be efficient in this or that extent, mainly for equations in one and two dimensions, from the end of the 1960s. Some of the principal and important examples given here are Toda systems of various types: abelian and nonabelian finite nonperiodic, periodic and affine Toda systems.