  We discuss novel generalisations of Killing tensors, which are introduced by
considering rational first integrals of geodesic equations. We introduce the
notion of inconstructible generalised Killing tensors, which cannot be
constructed from ordinary Killing tensors. Moreover, we introduce
inconstructible rational first integrals, which are constructed from
inconstructible generalised Killing tensors, and provide a method for checking
the inconstructibility of a rational first integral. Using the method, we show
that the rational first integral of the Collinson-O'Donnell solution is not
inconstructible. We also provide several examples of metrics admitting an
inconstructible rational first integral in two and four dimensions, by using
the Maciejewski-Przybylska system. Furthermore, we attempt to generalise other
hidden symmetries such as Killing-Yano tensors.
