  We determine the class of axisymmetric and asymptotically flat black-hole
spacetimes for which the test Klein-Gordon and Hamilton-Jacobi equations allow
for the separation of variables. The known Kerr, Kerr-Newman, Kerr-Sen and some
other black-hole metrics in various theories of gravity are within the class of
spacetimes described here. It is shown that although the black-hole metric in
the Einstein-dilaton-Gauss-Bonnet theory does not allow for the separation of
variables (at least in the considered coordinates), for a number of
applications it can be effectively approximated by a metric within the above
class. This gives us some hope that the class of spacetimes described here may
be not only generic for the known solutions allowing for the separation of
variables, but also a good approximation for a broader class of metrics, which
does not admit such separation. Finally, the generic form of the axisymmetric
metric is expanded in the radial direction in terms of the continued fractions
and the connection with other black-hole parametrizations is discussed.
