The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$, i.e., there are finite many polynomial triples $(x(t),y(t),z(t))$ with $x(t),y(t),z(t)\in\mathbb{Q}[t]$ satisfying the equation $x^3+y^3+z^3=2$.

**Question:** What might be the exact evidence for such an assertion?

**Edit:** Complementary materials on this problem:

Segre, Beniamino. "A note on arithmetical properties of cubic surfaces." *Journal of the London Mathematical Society* 1.1 (1943): 24-31.

Bremner, Andrew. "On diagonal cubic surfaces." *manuscripta mathematica* 62.1 (1988): 21-32.

The first paper states that a genus 0 curve on such diagonal cubic surface must be the complete intersection with another surface. The second one states that a genus 0 curve corresponding to a parametric solution should have some unusual properties at infinity. Although they are quite strong confinements, it is still ambiguous that why such a statement(finite many parametric solutions) tends to be reasonable.