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You can intersect with some hyperplanes to get a 0-dimensional variety, That means adding more polynomials to the ideal, say c1-4,c2-5,c3-6 (I haven't checked if this is 0-dimensional). You'll probably not find rational solutions that way, though, because the 0-dimensional variety that you do get probably doesn't have rational points.
In general, non-empty varieties can easily not have rational points. Take the variety defined by x^2+y^2-1 in Q[x,y]. Finding rational solutions on varieties is an unsolved problem.
Over an algebraically closed field, though, Hilbert's Nulstellensatz gives you that solutions exist and the hyperplane trick works. I'm not sure that .variety() is happy to return points in algebraically closed fields, though (in a way, the 0-dimensional ideal IS the algebraic point already, so there is not really a need to go further)
