Sums of two values of x cubed plus three x over the 3-adic integers ($n\equiv 0,\pm 1,\pm 4(\mathrm{mod}9)$)

For $n\in {\mathbb{Z}}_3$, the equation $x^3+3x+y^3+3y=n$ has a solution in the p-adic integers exactly for the displayed residue classes. The polynomial $F(t)=t^3+3t$ has values $0,4,5$ modulo nine, whose pairwise sums are $0,1,4,5,8$. For sufficiency choose initial pairs $(0,0),(-1,-1),(1,0),(-1,0),(1,1)$ for those five classes respectively, fix the second coordinate, and lift the first. Its derivative is $3(x^2+1)$, always of 3-adic norm $1/3$, while its second Taylor coefficient is $3x$, of norm below one. The residual is divisible by nine, so either the strict or the Boundary case of Hensel's lemma with a small quadratic coefficient applies.