I'm asking for a list of algebraic identities which "uniquely identify" (or nearly so) the fundamental solution (or those immediately around it) of a Pell equation.
As a concrete [numerical] example, the solutions to $X^2-2Y^2=\pm1$, alternately, are
$$(1,0),(1,1),(3,2),(7,5),(17,12), \dots$$
Observe that $(3,2),(7,5)$ is the only adjacent pair of solutions where the sum of the larger two elements is equal to the product of the smaller two, i.e., $7+3=10=2 \cdot 5$. So given adjacent solutions $(x_n,y_n)$ and $(x_{n+1},y_{n+1})$, we can state (and hopefully prove) that $$x_n+x_{n+1} = y_ny_{n+1} \iff (x_n,y_n)=(3,2).$$
Bonus points for algebraic proofs!