if r is the free associative algebra in two variables, say over the complex numbers, then the ring of two by two matrices over r is also a quotient of r by a differential ideal i. playing off the two descriptions of r/i leads naturally to the schwarzian korteweg-devries equation, in which a function of x evolves in time driven by the schwarzian derivative. i will present this abstract setup and then explain how ruled surfaces (appropriately chosen) over hyperelliptic curves provide solutions of the equation. i will also describe several commuting flows.