  Polynomial-in-time dependent symmetries are analysed for polynomial-in-time
dependent evolution equations. Graded Lie algebras, especially Virasoro
algebras, are used to construct nonlinear variable-coefficient evolution
equations, both in 1+1 dimensions and in 2+1 dimensions, which possess
higher-degree polynomial-in-time dependent symmetries. The theory also provides
a kind of new realisation of graded Lie algebras. Some illustrative examples
are given.
