On the integrability of homogeneous scalar evolution equations

We determine the existence of (infinitely many) symmetries for equations of the form

u(t) = u(k) + f(u, ..., u(k-1))

when they are lambda-homogeneous (with respect to the scaling u(k) lambda + k) with lambda > 0. Algorithms are given to determine whether a system has a symmetry (also independent of t and x). If it has one generalized symmetry, we prove it has infinitely many and these can be found using recursion operators or master symmetries. The method of proof uses the symbolic method and results From diophantine approximation theory. We list the 10 integrable hierarchies. The methods can in principle be applied to the lambda less than or equal to 0 cast. as we illustrate ibr one example with lambda = 0, which seems to be new. In principle they can also be used for systems of evolution equations, but so far this has only been demonstrated for one class of examples.