As in my previous answers, I will discuss only the solvability question: does a given equation have any integer solutions or not?
In my previous answers, I discussed the first item in the question: "One does not know the integral solutions of $P(x)=0$." All equations which belong to a well-known family of effectively solvable equations has been ignored. In that project, the current smallest open equation is $y(x^3-y)=z^3+3$ with $H=31$, see a separate Mathoverflow question Can you solve the listed smallest open Diophantine equations? for details.
In this answer, I will address the second item in the question: "There is a deterministic algorithm to find the integral solutions of $P(x)=0$, but the involved bounds are too big".
The current answers to this question are equations $$ x^3+x^2y-y^3-y+3 = 0 $$ and $$ y^3 = x^4+x+3 $$ with $H=29$.
The first equation has genus 1, and there is an effective upper bound for all potential integer solutions developed in [1]. The bound, however, is too large, despite some subsequent improvements. There is also an algorithm in [2], which is much faster but, to the best of my knowledge, was never implemented.
For the second equation, there is an effective upper bound for all potential integer solutions for any equation of the form $y^k=f(x)$, $k\geq 2$, under some minor conditions on polynomial $f(x)$, see [3], but the bound is too big. Much more promising is the effective Chabauty--Kim method, which is applicable for equations of genus $g\geq 2$ such that the rank $r$ of the Jacobian is less than the genus. For this equation, the genus $g=3$ and $r\leq 2$, hence the method should work in principle. For hyperelliptic equations, the method is actually implemented in Magma, but this equation is not hyperelliptic. Instead, it belongs to the family of Picard curves. For such equations, the case $r=0$ is resolved in the answer to this Mathoverflow question $y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves, while the case $r=1$ has been investigated in the recent work [4]. For our equation, however, the best known upper bound for the rank is $2$.
Of course, it is possible that these particular equations can be easily solvable by some elementary methods. If you solve any of them, please let me know in the comment, and I will update the answer with the next-smallest equations.
[1] Alan Baker and John Coates. Integer points on curves of genus 1. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 67, pages 595–602. Cambridge University Press, 1970.
[2] R. Stroeker, B.M.M de Weger, Solving Elliptic Diophantine Equations: The General Cubic Case, Acta Arith. 87 (4) (1998).
[3] Alan Baker. Bounds for the solutions of the hyperelliptic equation. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 65, pages 439–444. Cambridge University Press, 1969.
[4] Hashimoto, Sachi, and Travis Morrison. "Chabauty-Coleman computations on rank 1 Picard curves." arXiv preprint arXiv:2002.03291 (2020).
