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The structure of the sheath and the current exchange of two-dimensional electron-emitting objects with elliptic cross section immersed at rest in Maxwellian plasmas are investigated with an energy-conserving stationary Vlasov-Poisson solver free of statistical noise. The parameter domains for current collection within the orbital-motion-limited (OML) regime and current emission in space-charge-limited (SCL) conditions were studied by varying the characteristic dimension of the ellipse, its eccentricity, and the emission level. The analysis reveals the correlations between the onset of the non-OML and SCL regimes and the local curvature of the ellipse. As compared to nonemitting ellipses, electron emission broadens the parameter domain for OML current collection for ions and reduces considerably the current drop for non-OML conditions. Under identical plasma environments, elliptic bodies are more prone to operate under non-OML and SCL conditions than cylinders. Their emitted current in SCL conditions can be computed accurately from well-known results for cylinders if appropriate dimensionless variables and an equivalent radius are used. The role of the eccentricity, which acts as an integrability-breaking parameter, on the filamentation of the distribution function of the attracted species is studied.