The eclipsing polar CSS081231 turned bright (V_max ~ 14.5) in late 2008 and was subsequently observed intensively with small and medium-sized telescopes. A homogeneous analysis of this comprehensive dataset comprising 109 eclipse epochs is presented and a linear ephemeris covering the five years of observations, about 24000 orbital cycles, is derived. Formally this sets rather tight constraints on the mass of a hypothetical circumbinary planet, M_pl <= 2 M_Jup. This preliminary result needs consolidation by long-term monitoring of the source. The eclipse lasts 433.08 +- 0.65 s, and the orbital inclination is found to be i=79.3 - 83.7 degrees. The centre of the bright phase displays accretion-rate dependent azimuthal shifts. No accretion geometry is found that explains all observational constraints, suggesting a complex accretion geometry with possible pole switches and a likely non-dipolar field geometry.
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