(a) Do there exist integers $x$ and $y$ such that $x^3+x^2y^2+y^3=7$ ?


(b) Is this equation belongs to some family $F$ of equations for which there is a known algorithms for testing if they have an integer solution? For example, such algorithms are known for quadratic equations (in $n$ variables) and for cubic equations in 2 variables, but this equation has degree 4.


(c) Reference request: can you recommend a good book/survey/website which would help to determine if a given (reasonably simple) Diophantine equation belong to some solved case, or is new?



$x^3+x^2y^2+y^3=7$, and solvable families of Diophantine equations
(a) No. There are no integer solutions. The curve $C$ you give has genus $3$ and it has an obvious automorphism $\phi(x,y) = (y,x)$. The quotient curve is an elliptic curve. In particular, if you let $X = -(x+y)$ and $Y = xy$, then the equation becomes $E : Y^{2} + 3XY = X^{3} + 7$. So any integer point on your curve gives rise to an integer point on $E$. As you note, cubic equations in two variables have algorithms for finding their integer points. This elliptic curve has rank $1$ and its integral points are $(-3,4)$, $(-3,5)$, $(186,-2831)$ and $(186,2273)$. None of these lead to integer points on $C$.


(b) There are not many general families of such curves. For curves of genus $0$, there are algorithms for finding their integer points. For curves of genus $1$, the existing procedures rely on being able to compute the Mordell-Weil group and it is currently open whether there is an algorithm to do so. One other family that might be worth mentioned is the family of [Thue equations](https://en.wikipedia.org/wiki/Thue_equation), those of the form $F(x,y) = k$ where $k$ is a constant and $F$ is a homogeneous polynomial in two variables.


(c) For a survey of what number theorists can and cannot handle algorithmically, you might consult Henri Cohen's book "Number Theory, Volume I: Tools and Diophantine Equations." Chapter 6 in that book gives a nice survey of simple Diophantine equations together with some techniques that might suffice to handle those. The class of equations that can be definitely be handled in a systematic or algorithmic way is quite small. (If I had been unable to compute the Mordell-Weil group of $E$ above, the software would not have been able to provably find all the integral points.)


