  We address the problem of how to characterise when a rank-two conformal
Killing tensor is the trace-free part of a Killing tensor for a metric in the
conformal class. We call such a metric a Killing scale. Our approach is via
differential prolongation using conformally invariant tractor calculus. First,
we show that there is a useful partial prolongation of the conformal Killing
equation to a simplified equation for sections of some tractor bundle. We then
use this partial prolongation to provide such an invariant characterisation in
terms of the scale tractor and this partial prolongation. This captures
invariantly the relevant Bertrand--Darboux equation. We show that Einstein
Killing scales have a special place in the theory. On conformally flat
manifolds, we give the full prolongation of the conformally Killing equation to
a conformally invariant connection on a tractor bundle. Using this, we provide
a characterisation of (non-scalar flat) Einstein Killing scales by an algebraic
equation for the scale tractors corresponding to such metrics. This also
provides an algebraic description of the linear subspace of conformal Killing
tensors that are compatible with a given Einstein Killing scale. For
completeness and to introduce the main ideas, we also study analogous questions
for conformal Killing vectors.
