Identifying Ramanujan's integer solutions of x^3+y^3+z^3=1 among Elkies' rational solutions 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?
 A: I am not an expert here, so please take this as a remark. But I think, Ramanujan's solution 
appears to hinge on special circumstances of the solution
$$
(x^2 +7xy−9y^2)^3 +(2x^2 −4xy+12y^2)^3 = (2x^2 +10y^2)^3 +(x^2 −9xy−y^2)^3,
$$
which is claimed in http://thales.math.uqam.ca/~rowland/papers/Known_families_of_integer_solutions_of_x%5E3+y%5E3+z%5E3=n.pdf. 
On the site sites.google.com/site/tpiezas/010, the following identity is also attributed to
Ramanujan:
$$
(3x^2+5xy-5y^2)^3 + (4x^2-4xy+6y^2)^3 + (5x^2-5xy-3y^2)^3 = (6x^2-4xy+4y^2)^3,
$$
Edit: However, this cannot give solutions for $x^3+y^3+z^3=1$, see Michael's comment.
Piezas also has found a way to generate other Ramanujan-like families of solutions.
See also: https://math.stackexchange.com/questions/381111/generalizing-ramanujans-sum-of-cubes-identity
A: I am not saying anything you must not already know, but you can solve the equations in $(r,s,t)$ for the Ramanujan solutions of small $n$.
$(1,2,-2)\rightarrow (0,-1,2)$
$(135,138,-172)\rightarrow (-238,-299,388)$
$(11161,11468,-14258)\rightarrow (-6510,-8269,10742)$
$(926271,951690,-1183258)\rightarrow (-1620976,-2058689,2674342)$
$(76869289,78978818,-98196140)\rightarrow (-44840418,-56948791,73979396)$
