What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis?

Please see this related post and also the following post.. For Mathematical development around this historical problem please see this paper.

**Added :** According to their method, what of the following two statements are true?:

There are uniform numbers $\tilde{H}(n)$ such that every polynomial vector field $X$ of degree $n$ satisfies:

**Statement 1)** There are at most $\tilde{H}(n)$ real limit cycles of $X$ which lie on the same leaf.

**statement 2)** There are at most $\tilde{H}(n)$ distinct complex leaves which contains real limit cycles.

By "Leaf" I mean the leaf of the corresponding complex singular foliation of $\mathbb{C}P^{2}$. Some technical and historical aspects of these foliations are explained here. However in this linked paper there is no an explicit explianation about the "error".

According to the video of lecture of Ilyashenko, provided in the answer to this question by Andrey Gogolev, we ask:

What is the fate of the "persistence problem" which is mentioned by Ilyashenko? How can it be revised to become a true statement?

According to the first page of the english version of the paper of Petrovski_Landis we ask

"How they assume that a solution of the equation can be considered as an

entiremap from $\mathbb{C}$ to $\mathbb{C}P^{2}$? Can every leaf be parametrized by an entire map? Does this "entire-assumption" play an important role in their proof? Please See this related post.

According to comments and answers to this question, we undrestand there is no a **written paper** which explains the **error**, explicitly.**Why really this is the case?**