  Ricci-flat spacetimes of signature (2,q) with q=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.
