Integrable systems are mathematical models that can be solved exactly, often yielding a complete set of solutions. These systems typically exhibit a high degree of symmetry and conservation laws, making them easier to analyze. Examples include the pendulum and Kepler's problem, where the motion can be predicted over time. In contrast to chaotic systems, integrable systems allow for predictable behavior, often described using Hamiltonian mechanics. They play a crucial role in various fields, including classical mechanics, quantum mechanics, and mathematical physics, providing insights into the underlying structure of complex dynamical systems.