(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?