The motivation of the following question comes from the Problem D4 in the book "unsolved problems in number theory" by Richard Guy.
No integral points on the surface $S:x^3+y^3+z^3=3$ is known other than $(1,1,1),(4,4,-5)$ and their permutations. Although integral points are sparse on the surface, it is well known that we can construct a large number of rational points from a given rational point $(a,b,c)$. Any rational line passing through $(a,b,c)$ must intersect $S$ at two points $P,\bar{P}$, which are conjugate to each other over some quadratic field $K$. The tangent plane passing through $P$($\bar{P}$) intersects S at some rational plane curve $C$ over $K$. Join any point on $C$ over $K$ and its conjugate by line, and it will intersect $S$ at a rational point.
Question: The procedure constructs a family of rational points. Is there any way to find out an integral point in such families?