  We examine the separability properties of the equation of motion for a
stationary string near a rotating charged black hole with two independent
angular momenta in five-dimensional minimal gauged supergravity. It is known
that the separability problem for the stationary string in a general stationary
spacetime is reduced to that for the usual Hamilton-Jacobi equation for
geodesics of its quotient space with one dimension fewer. Using this fact, we
show that the "effective metric" of the quotient space does not allow the
complete separability for the Hamilton-Jacobi equation, albeit such a
separability occurs in the original spacetime of the black hole. We also show
that only for two special cases of interest the Hamilton-Jacobi equation admits
the complete separation of variables and therefore the integrability for the
stationary string motion in the original background: namely, when the black
hole has zero electric charge or it has an arbitrary electric charge, but two
equal angular momenta. We give the explicit expressions for the Killing tensors
corresponding to these cases. However, for the general black hole spacetime the
effective metric of the quotient space admits a conformal Killing tensor. We
construct the explicit expression for this tensor.
