  The exact solution for the motion of a test particle in a non-spherical polar
orbit around a Kerr black hole is derived. Exact novel expressions for frame
dragging (Lense-Thirring effect), periapsis advance and the orbital period are
produced. The resulting formulae, are expressed in terms of Appell's first
hypergeometric function $F_1$, Jacobi's amplitude function, and Appell's $F_1$
and Gau$\ss$ hypergeometric function respectively. The exact expression for
frame dragging is applied for the calculation of the Lense-Thirring effect for
the orbits of S-stars in the central arcsecond of our Galaxy assuming that the
galactic centre is a Kerr black hole, for various values of the Kerr parameter
including those supported by recent observations. In addition, we apply our
solutions for the calculation of frame dragging and periapsis advance for
stellar non-spherical polar orbits in regions of strong gravitational field
close to the event horizon of the galactic black hole, e.g. for orbits in the
central milliarcsecond of our galaxy. Such orbits are the target of the GRAVITY
experiment. We provide examples with orbital periods in the range of 100min -
54 days. Detection of such stellar orbits will allow the possibility of
measuring the relativistic effect of periapsis advance with high precision at
the strong field realm of general relativity. Further, an exact expression for
the orbital period of a test particle in a non-circular equatorial motion
around a Kerr black hole is produced. We also derive exact expressions for the
periapsis advance and the orbital period for a test particle in a non-circular
equatorial motion in the Kerr field in the presence of the cosmological
constant in terms of Lauricella's fourth hypergeometric function $F_D$.
