Noah Schweber gives an excellent account of the context on MO, so I will only add the missing piece concerning the error Ilyashenko and Novikov found in the original Landis-Petrovskiĭ's argument. They never published it, but Landis and Petrovskiĭ wrote a letter to the editors (1967) of Mathematics of the USSR-Sbornik, which published their original paper in 1955. It reads as follows:
"In our paper On the number of Limit Cycles of the Equation $dy/dx=P(x,y)/Q(x,y)$, where $P$ and $Q$ are Polynomials of the Second Degree in the proof of lemma 12 (p.242) there is an error pointed out to us by S.P. Novikov. A reference to this lemma is made in our [follow up] paper On the number of Limit Cycles of the Equation $dy/dx=P(x,y)/Q(x,y)$, where $P$ and $Q$ are Polynomials. Therefore, one can only claim that the results of those works hold under the additional assumption of that lemma's validity".
Unfortunately, both original papers are from before 1967, when AMS started making English translations of the Sbornik, but they link to five (!) papers by Ilyashenko starting in 1969 that cite them, meaning their translations are available. At least to those who have the IOP Science subscription... unlike myself. Three of them are from 1969, one is titled An Example of Equations $dw/dz=P_n(z,w)Q_n(z,w)$ having a Countable Number of limit Cycles and Arbitrarily Large Petrovskii–Landis Genus, so they may have more details on the error.