On space--time Killing tensors with a Segre characteristic [(11)(11)]

We consider the set of all space--times which admit a Killing tensor K/sub αβ/ whose Segre characteristic is [(11)(11)] and whose eigenvalues lambda1 and lambda2 satisfy the condition that dlambda1 X dlambda2 is not null. We prove that there locally exist scalar fields phi3, phi4 such that lambda1, lambda2, phi3, phi4 constitute a chart, and dlambda1 x dlambda2 = dlambda/sup i/ x d phi/sup r/ = 0 for i=1, 2 and r = 3, 4. Also, relative to this coordinate system, the metric has the same general form as Carter's Hamilton--Jacobi separable metric except that his arbitrary functions of lambda/sup i/ are replaced by functions of lambda/sup i/, phi3, phi4. If R/sup αβ/(del/sub α/lambda1)(del/sub β/lambda2) = 0, there exists a two-parameter Abelian group of isometries which commute with K/sub αβ/; also /phi3,phi4) can be chosen so that par.delta par.delta phi3 and par.delta/par.delta phi4 are the Killing vectors. R/sup αβ/(del/sub α/lambda1)(del sub β/lambda2) = 0 is necessary and sufficient for Schroedinger equation separability in case (a) of Carter. A Newtonian analog of our results is discussed