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24 pages, no figuresInternational audienceIn this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincaré--Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are also detectable with the Poincaré--Liapunov method

Let X(x, y) and Y (x, y) be real analytic functions without constant and linear terms defined in a neighborhood of the origin. Assume that the analytic differential systeṁ x = y + X(x, y),ẏ = Y (x, y) has a nilpotent center at the origin. The first integrals, formal or analytic, will be real except if we say explicitly the converse. We prove the following. (a) If X = yf (x, y 2 ) and Y = g(x, y 2 ), then the system has a local analytic first integral of the form H = y 2 + F (x, y), where F starts with terms of order higher than two. (b) If the system has a formal first integral, then it has a formal first integral of the form H = y 2 + F (x, y), where F starts with terms of order higher than two. In particular, if the system has a local analytic first integral defined at the origin, then it has a local analytic first integral of the form H = y 2 + F (x, y), where F starts with terms of order higher than two. (c) As an application we characterize the nilpotent centers for the differential systemṡ x = y + P 3 (x, y),ẏ = Q 3 (x, y), which have a local analytic first integral, where P 3 and Q 3 are homogeneous polynomials of degree three.

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