Abstract
We construct a polynomial differential system that admits a given set of invariant algebraic curves. For such a system we solve the Darboux problem (the existence of the Darboux first integral), the Poincare problem (the existence of an upper bound for the degree of invariant algebraic curve) and study Hilbert's 16th problem for algebraic limit cycles (the existence of an upper bound for the number of algebraic limit cycles).
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