  We review some essential aspects of classically integrable systems. The
detailed outline of the lectures consists of: 1. Introduction and motivation,
with historical remarks; 2. Liouville theorem and action-angle variables, with
examples (harmonic oscillator, Kepler problem); 3. Algebraic tools: Lax pairs,
monodromy and transfer matrices, classical r-matrices and exchange relations,
non-ultralocal Poisson brackets, with examples (non-linear Schroedinger model,
principal chiral field); 4. Features of classical r-matrices: Belavin-Drinfeld
theorems, analyticity properties, and lift of the classical structures to
quantum groups; 5. Classical inverse scattering method to solve integrable
differential equations: soliton solutions, spectral properties and the
Gel'fand-Levitan-Marchenko equation, with examples (KdV equation, Sine-Gordon
model). Prepared for the Durham Young Researchers Integrability School,
organised by the GATIS network. This is part of a collection of lecture notes.
