Given,

$$a^4+b^4+c^4 = d^4\tag{0}$$

we have the identity,

$$(-11980 + 1673 u + 54u^2)^4 + (36 - 2321 u + 3u^2)^4 + t^4 = (24677 + 203 u + 71u^2)^4$$

where,

$$591800025 + 20030510 u + 1671327 u^2 + 92762 u^3 - 4112 u^4 = t^2\tag{1}$$

as well as a second,

$$(62697 + 5045 v - 242v^2)^4 + (-19200 - 9089 v + 46v^2)^4 + t^4 = (86825 - 27 v + 303v^2)^4$$

where,

$$-6422010512 + 412760610 v - 6214161 v^2 + 2027190 v^3 + 70673 v^4 = t^2\tag{2}$$

I know of only one rational solution of *small height* each to $(1)$ and $(2)$, namely,

$$ u =-2020/127$$

$$ v = -8251/94$$

Thus $(1)$ and $(2)$ can be transformed into *elliptic curves*, but they are *distinct* from the one used by Elkies to find the first solution to $(0)$, or the one that yields the smallest solution (found by R. Frye). From the initial rational point, I know how to generate others, but the numerators and denominators are huge.

** Question**: Anybody has software to find other rational solutions

*of small height*to $(1)$ and $(2)$ ?

** P.S.** This question is related to the one I asked in MSE.

ratpointsfor about a minute on one head of the Sage cluster, I find that the first curve also has a pair of rational points at $u = 76164/2063$, and this is the only further point on either curve up to height $5 \cdot 10^5$. $\endgroup$ – Noam D. Elkies Oct 2 '13 at 2:42