In general, it is not clear What does it mean to solve an equation? in integers. In this question, let us assume that an equation $$ P(x_1,\dots,x_n)=0 $$ is solved if we have proved that its integer solution set $S \subset {\mathbb Z}^n$ can be represented as the finite union $S=S_1 \cup \dots \cup S_m$, where each $S_i$ is either a polynomial family or a family defined by recurrence relations. Here, $S \subset {\mathbb Z}^n$ is a polynomial family if there exists polynomials $P_1,\dots,P_n$ in some variables $u_1,\dots,u_k$ and integer coefficients such that $(x_1,\dots,x_n) \in S$ if and only if there exists integers $u_1,\dots,u_k$ such that $x_i=P_i(u_1,\dots,u_k)$ for $i=1,\dots,n$.

Following Zidane What is the smallest unsolved Diophantine equation? , let us define size $H$ of the equation $P=0$ as a result of substitution 2 instead of all variables, absolute values instead of all coefficients, and evaluating.

All equations with $H \leq 8$ are easy to solve. However, simple-looking equation $xy-zt=1$ with $H=9$ has been open for decades. In 2010, Vaserstein¹ https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n2-p07-p.pdf proved that

• The solution set to $xy-zt=1$ is a polynomial family with $46$ parameters.

As a corollary of this, Vaserstein solved the following families of equations

• $xy-zt=D$ for any integer $D$;
• $yz=x^2+D$ for any integer $D$;
• $x_1x_2+x_3x_4+Q(x_5,\dots,x_n)=D$ for quadratic form $Q$ and integer $D$.

In addition, the following equations/families has been solved:

• Equations in the form $dyz=ax^2+bx+c$ for integers $a,b,c,d$ has been solved in the answer to question Polynomial parametrization of the solutions to $yz=x^2+x\pm 1$
• All equations with $H \leq 12$.

I was also able to solve all equations with $H=13$ except of the following ones. $$ x^3 + 1 = yz $$ $$ x^2y=z^2 \pm 1 $$ $$ x^2y=tz+1 $$ $$ x^2 + y^2 \pm 1 = zt $$ $$ x^2 \pm 1 = yzt $$ $$ x_1x_2x_3+x_4x_5=1 $$

For each of the listed equations, the question is to find all integer solutions. Specifically, check whether the set of all integer solutions is a finite union of polynomial families and/or families defined by recurrence relations. You do not need to write the resulting families explicitly, because, as example $xy-zt=1$ indicates, they may be quite complicated.

See here Can you solve the listed smallest open Diophantine equations? for a version of this question where we only want to check whether any integer solution exists, and here On the smallest open Diophantine equations: beyond Hilbert's 10 problem for a version where we also check whether the solution set is finite or infinite (and find all solutions if there are finitely many).

¹Vaserstein, Leonid, Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups, Ann. Math. (2) 171, No. 2, 979-1009 (2010). ZBL1221.11082, JSTOR.