Completely integrable system

A completely integrable system is like a special puzzle that has already been solved. Imagine that you have a bunch of shapes that you need to fit into a box. A regular puzzle might be really hard to solve because the pieces don't fit together perfectly. But if you have a completely integrable puzzle, someone has already figured out how to put all the pieces in the box so they fit together perfectly.

In math, a completely integrable system is a kind of equation that has already been solved. It's like someone has already figured out the answer to the equation, so you don't need to work hard to solve it yourself. This can be really useful in physics, because many physical systems can be described by equations that are really hard to solve. But if you can find a completely integrable system that describes your physical system, you can just use the solution that already exists instead of trying to solve it yourself. It's like having a shortcut to the answer.