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In this Post I derive the differential equations of motion of a homogeneous elliptic lamina of mass m and the major and minor axes of lengths of a and b which rolls without slipping along the horizontal x axis within the vertical xy plane.

If the initial angular velocity is large enough, the ellipse will roll forever and go to ±∞ in the x direction, otherwise it will just rock.

I have attached two files:

 rolling-ellipse.mw
        Worksheet to solve the differential equations and animate the motion

rolling-ellipse.pdf
         Documentation containing the derivation of the differential equations

And here are two animations extracted from the worksheet.
differential-equation animation ellipse roll

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