A symmetry approach to exactly solvable evolution equations

A method is developed for establishing the exact solvability of nonlinear evolution equations in one space dimension which are linear with constant coefficient in the highest-order derivative. The method, based on the symmetry structure of the equations, is applied to second-order equations and then to third-order equations which do not contain a second-order derivative. In those cases the most general exactly solvable nonlinear equations turn out to be the Burgers equation and a new third-order evolution equation which contains the Korteweg-de Vries (KdV) equation and the modified KdV equation as particular cases