$x^3+x^2y^2+y^3=7$, and solvable families of Diophantine equations (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?
 A: Let me post an elementary answer to this question, which I have found recently.
(a) Assume that $(x,y)$ is an integer solution. From symmetry, we may assume that $|y|\geq |x|$. If fact, cases $|y|=|x|$, $|y|=|x|+1$, and $|x|\leq 3$ can be checked directly, hence we may assume that $|y|\geq |x|+2>5$.
Now rewrite the equation as
$$
(x^2+y)(y^2+x)=xy+7,
$$
from which it follows that
$$
k = \frac{xy+7}{y^2+x}
$$
is an integer. However,
$$
y^2+x \geq |y|(|x|+2)-|x|=|xy|+2|y|-(|y|-2)=|xy|+|y|+2>|xy|+7\geq |xy+7|
$$
hence $|k|<1$, and it can be an integer only if $k=0$, or $xy+7=0$, which is impossible for $|y|\geq |x|+2>5$.
(b) This equation belongs to family of equations satisfying Runge's condition, for which there exists a general algorithms, see, for example
Walsh, P. A quantitative version of Runge's theorem on Diophantine equations, Acta Arithmetica, 62, 157--172, 1992.
A: (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, 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.)
