  The Kustaanheimo-Stiefel transformation of the Kepler problem with a
time-dependent perturbation converts it into a perturbed isotropic oscillator
of 4-and-a-half degrees of freedom with additional constraint known as bilinear
invariant. Appropriate action-angle variables for the constrained oscillator
are required to apply canonical perturbation techniques in the perturbed
problem. The Lissajous-Kustaanheimo-Stiefel (LKS) transformation is proposed,
leading to the action-angle set which is free from singularities of the LCF
variables earlier proposed by Zhao. One of the actions is the bilinear
invariant, which allows the reduction back to the 3-and-a-half degrees of
freedom. The transformation avoids any reference to the notion of the orbital
plane, which allowed to obtain the angles properly defined not only for most of
the circular or equatorial orbits, but also for the degenerate, rectilinear
ellipses. The Lidov-Kozai problem is analyzed in terms of the LKS variables,
which allow a direct study of stability for all equilibria except the circular
equatorial and the polar radial orbits.
