Section 8.3.2 of recent book [1] studies the following problem. Define the length of a Polynomial Diophantine equation as the sum of degrees of monomials plus sum of base 2 logarithms of the coefficients, order the equations by length, and investigate the **existence** of integer solutions (you do not need to find all solutions). The shortest cubic equation left open in the book is
$$
1 + 3 x^3 + x y^2 + 6 y z^2 = 0  \quad\quad\quad (1)
$$
The question is whether there exist integers $x,y,z$ satisfying this equation. With $t=xy$, this reduces to the solvability of $x+3x^4+t^2+6tz^2=0$ such that $x$ is a divisor of $t$.
 
**Update 20.09.2024:** Dmitry Ezhov now found some solutions to (1) starting from $(x,y,z)
=(-1017461719,95574914,2350866170)$, see comments. So, this equation is now resolved. The next-shortest open cubic equation in [1] is  
$$
y^2+10xyz+x^3-x-2=0.  \quad\quad\quad (2)
$$
Modulo $20$ analysis shows that $x=2$ mod $20$. After replacing $y$ with $-y$ if necessary, we may also assume that $y=4$ mod $10$.

**Update 07.10.2024:** Dmitry Ezhov now found a solution
$$
(23499130751021842,252697047241468990409432149765008357514,-1075346360334969622883)
$$
to (2). A solution of this size would be unfeasible to find by direct search, but nice ascent method for the related equation $y^2+xyz+x^3-x-2=0$ has been used, see comments.

The next-shortest open cubic equation in [1] is  
$$
y^2+7xyz+3x^3-2=0.  \quad\quad\quad (3)
$$
Modulo $4$ and $7$ analysis shows that $x=7$ mod $14$. After replacing $y$ with $-y$ if necessary, we may also assume that $y=3$ mod $14$. Does this equation have any integer solutions? Can a solution be found by first developing ascent formulas for the solutions to $y^2+xyz+3x^3-2=0$?

[1] Bogdan Grechuk, Polynomial Diophantine equations. A systematic approach, Springer, 2024