In his Lost Notebook, Ramanujan exhibits infinitely many integer solutions to $x^3+y^3+z^3=1$. On his webpage (http://www.math.harvard.edu/~elkies/4cubes.html), Elkies determines all rational solutions to this equation. I have two questions: does Elkies' result enable one to determine all integer solutions, and does Elkies' result at least enable one to recover Ramanujan's integer solutions?

Ramanujan's integer solutions are as follows: write $$ \frac{1+53t+9t^2}{1-82t-82t^2+t^3} = x_0 + x_1t + x_2t^2 + ... $$ $$ \frac{2-26t-12t^2}{1-82t-82t^2+t^3} = y_0 + z_1t + y_2t^2 + ... $$ $$ \frac{-2-8t+10t^2}{1-82t-82t^2+t^3} = z_0 + z_1t + z_2t^2 + ... $$ Then, for every $n$, it is true that $$ x_n^3 + y_n^3 + z_n^3 = (-1)^n. $$ (For a proof, see J.H.Han and M.D.Hirschhorn, "Another look at an amazing identity of Ramanujan", Math.Mag. 79 (2006), 302-304, or alternately either of two earlier papers by Hirschhorn cited in that one.)

Elkies showed that all rational solutions to $x^3+y^3+z^3=1$ can be written as $(x,y,z)=(\frac AD,\frac BD,\frac CD)$ where there are integers $r,s,t$ for which $$ A=-(s+r)t^2 + (s^2+2r^2)t - s^3 + rs^2 - 2r^2s - r^3 $$ $$ B = t^3-(s+r)t^2+(s^2+2r^2)t+rs^2-2r^2s+r^3 $$ $$ C = -t^3+(s+r)t^2-(s^2+2r^2)t+2rs^2-r^2s+2r^3 $$ $$ D=(2r-s)t^2+(s^2-r^2)t-s^3+rs^2-2r^2s+2r^3. $$

Does anyone have an idea of how to connect these two results?