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.