While working on another problem (https://mathoverflow.net/questions/143599/solving-the-quartic-equation-r4-4r3s-6r2s2-4rs3-s4-1), I came across a question which seems to be of [semi-] independent interest.

**Conjecture.** If $r > s \ge 1$ are integers such that $r$ is odd and $s$ is even and
$$(r+s)^2 \mid (4r^4+1), \qquad(\star)$$
then $(r,s) \in \{(3,2), (21,8), (119,50), (697,288), (2679,910), (4059,1682), \dots\}$.

Note that these are essentially Pell oblongs (_i.e._, $r$ is http://oeis.org/A001652) and doubles of Pell squares (_i.e._, $s$ is http://oeis.org/A114619), **except for** the outlier solution $(2679,910)$, which is the only one maxima gave for $r \le 11000$ (I'm increasing that bound now).

I haven't the foggiest notion how to prove any conditions on $r$ and $s$ satisfying $(\star)$, or to figure out why there are any 'outlier' solutions at all (and, of course, if there are any more). EDIT: For example, it seems that all solutions (including the 'outlier') with $r>3$ satisfy $s/r < 1/2$; can one prove this bound?

Any suggestions, hints, full solutions, _etc._ greatly appreciated.