Graded Symmetry Algebras of Time-Dependent Evolution Equations and Application to the Modified KP equations

By starting from known graded Lie algebras, including Virasoro algebras, new kinds of time-dependent evolution equations are found possessing graded symmetry algebras. The modified KP equations are taken as an illustrative example: new modified KP equations with $m$ arbitrary time-dependent coefficients are obtained possessing symmetries involving $m$ arbitrary functions of time. A particular graded symmetry algebra for the modified KP equations is derived in this connection homomorphic to the Virasoro algebras.