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In [Kim05], Kim gave a new proof of Siegel's Theorem that there are onlyfinitely many $S$-integral points on $\mathbb P^1_{\mathbbZ}\setminus\{0,1,\infty\}$. One advantage of Kim's method is that it inprinciple allows one to actually find these points, but the calculations growvastly more complicated as the size of $S$ increases. In this paper, weimplement a refinement of Kim's method to explicitly compute various exampleswhere $S$ has size $2$ which has been introduced in [BD19]. In so doing, weexhibit new examples of a natural generalisation of a conjecture of Kim.

Refined Selmer equations for the thrice-punctured line in depth two