Ricci-flat spacetimes admitting higher rank Killing tensors

Ricci-flat spacetimes of signature (2,q)(2,q) with q=2,3,4q=2,3,4 are constructed which admit irreducible Killing tensors of rank-3 or rank-4. The construction relies upon the Eisenhart lift applied to Drach's two-dimensional integrable systems which is followed by the oxidation with respect to free parameters. In four dimensions, some of our solutions are anti-self-dual.