  We derive Keplerian-type parametrization for the solution of post-Newtonian
(PN) accurate conservative dynamics of spinning compact binaries moving in
eccentric orbits. The PN accurate dynamics that we consider consists of the
third post-Newtonian accurate conservative orbital dynamics influenced by the
leading order spin effects, namely the leading order spin-orbit interactions.
The orbital elements of the representation are explicitly given in terms of the
conserved orbital energy, angular momentum and a quantity that characterizes
the leading order spin-orbit interactions in Arnowitt, Deser, and Misner-type
coordinates. Our parametric solution is applicable in the following two
distinct cases: (i) the binary consists of equal mass compact objects, having
two arbitrary spins, and (ii) the binary consists of compact objects of
arbitrary mass, where only one of them is spinning with an arbitrary spin. As
an application of our parametrization, we present gravitational wave
polarizations, whose amplitudes are restricted to the leading quadrupolar
order, suitable to describe gravitational radiation from spinning compact
binaries moving in eccentric orbits. The present parametrization will be
required to construct `ready to use' reference templates for gravitational
waves from spinning compact binaries in inspiralling eccentric orbits. Our
parametric solution for the post-Newtonian accurate conservative dynamics of
spinning compact binaries clearly indicates, for the cases considered, the
absence of chaos in these systems. Finally, we note that our parametrization
provides the first step in deriving a fully second post-Newtonian accurate
`timing formula', that may be useful for the radio observations of relativistic
binary pulsars like J0737-3039.
