Let $Q(x,y,z)$ be a cubic polynomial with integer coefficients, such that the terms $x^3, y^3, z^3$ do not appear. That is, it is at most quadratic in each of the variables $x,y,z$.


Is there a general method to count integral points $(a,b,c)$ with $\max\{|a|, |b|, |c|\} \leq T$ on the affine cubic surface defined by $Q(x,y,z) = 0$?


The prototypical example is the Markoff surface defined by $Q(x,y,z) = x^2 + y^2 + z^2 - xyz$. Here Zagier showed that the density of integral points is asymptotic to $C (\log T)^2$ for some explicit constant $C > 0$. This is generalized by Baragar and Umeda in [this paper](https://www.cambridge.org/core/journals/bulletin-of-the-australian-mathematical-society/article/the-asymptotic-growth-of-integer-solutions-to-the-rosenberger-equations/20AC92677B51DCF5ACA52AB8FF52DC9F).


Their method depends on an explicit descent on the Markoff surface, namely that for a given point $(a,b,c)$ the point $(bc - a, b, c)$ is also a point. Zagier then showed that all integral solutions are generated from the fundamental solution $(3,3,3)$ and by permuting the variables and applying the above map. This means that the set of solutions grows exponentially in size, hence giving the bound $O((\log T)^2)$.


Is there something similar that can be done, using only the fact that all of the monomials in $Q$ are at most quadratic in each variable?



Density of integral points on affine cubic surfaces of a certain type
Not in general.


The involutions of the Markov surface such as
$(a,b,c) \leftrightarrow (bc-a,b,c)$ preserve integral points because
$x^2 + y^2 + z^2 - xyz$ is a *monic* quadratic polynomial in each variable.
That works more generally for any polynomial of the form
$Q(x,y,z) = x^2 + y^2 + z^2 - L(x,y,z)$ with $L$ linear in each variable,
though already in this special case $Q=0$ can have infinitely many solutions
(it doesn't even have to be irreducible, e.g. $(x \pm y \pm z)^2$ works).
For the general cubic polynomial with no $x^3,y^3,z^3$ terms,
there are still rational involutions that fix three of the variables,
but they have denominators and thus need not send integral points to
integral points (and occasionally will send a rational or even integral point
to infinity, when the quadratic coefficient w.r.t. one of the variables
vanishes).


