<s>[INST] A caboodle of Pell's equation in one? $x^2+y^2-5xy+5=0$ I saw this twitter post that reads:

Find all the pairs of positive integers $(x,y)$ satisfying $$  x^2 + y^2 - 5xy + 5 = 0 . $$

I don't know how to tackle this and I ended up summoning WolframAlpha which shows that there are infinitely many solutions. What's interesting is that there are (at least?) 12 general forms that looks like they all resemble the solutions to Pell's equation.
For example, the first general solution presented reads:

$$ \begin{array} { r c l }
 x &=& 
\dfrac1{42} \Big [ 21 \left(55 - 12\sqrt{21} \right)^n 
- \sqrt{21} \left(55 - 12\sqrt{21} \right)^n 
+ 21 \left(55 + 12\sqrt{21} \right)^n 
+ \sqrt{21} \left(55 + 12\sqrt{21} \right)^n \Big ] \\ \phantom0 \\
 y &=& 
\dfrac1{42} \Big [ 63 \left(55 - 12\sqrt{21} \right)^n 
- 13 \sqrt{21} \left(55 - 12\sqrt{21} \right)^n 
+ 63 \left(55 + 12\sqrt{21} \right)^n 
+ 13 \sqrt{21} \left(55 + 12\sqrt{21} \right)^n \Big] \end{array} $$ for $n = 0,1,2,3\ldots $.

Plugging $n=0,1,2,3$ gives the first few solutions
$$(x,y) = (1,3), (67,321), (7369,35307), (810523,3883449) 
$$
I tried converting these solutions to the form of $X^2 - DY^2 = A$ or something similar to Pell's equation but I got nothing.
Is it a coincidence that this innocuous-looking quadratic Diophantine equation is actually a caboodle of Pell's equation in disguise? If so, how can we derive them all?
 [/INST] This sort of thing has a clear description in terms of Conway's topograph; I find it more convenient to use equivalent form $u^2 + uv - 5 v^2.$  The outcome for the original problem is sequences (note that the rule deals with every other element). For instance,
$5\cdot 9 - 2 = 43$  and $5 \cdot 14 -3 = 67$
$$   x_{n+4}= 5 x_{n+2} - x_n,$$
$$   y_{n+4}= 5 y_{n+2} - y_n,$$
There are two interleaved subsequences.
$$
\begin{array}{c}
1&1&2&3&9&14&43&67&206&321&987 &1538&4729& \ldots \\
3&2&1&1&2&3&9&14&43&67&206&321&987&\ldots \\
\end{array}
$$
Running the indices backwards leads to different solutions, but they are just transpositions of the ones above.
Let's see, given $u^2 + uv - 5 v^2 = -5$ and $x=u+3v, y=v$ gives $x^2 - 5 xy + y^2 = -5$ and the reverse holds as well

Apparently I drew one of these in 2016

Examples in previous answers:
Generate solutions of Quadratic Diophantine Equation
diagrams
Another quadratic Diophantine equation: How do I proceed?
How to find solutions of $x^2-3y^2=-2$?
Generate solutions of Quadratic Diophantine Equation
Why can't the Alpertron solve this Pell-like equation?
Finding all solutions of the Pell-type equation $x^2-5y^2 = -4$
If $(m,n)\in\mathbb Z_+^2$ satisfies $3m^2+m = 4n^2+n$ then $(m-n)$ is a perfect square.
how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $ (x,y)$  ::::  69   55
Find all integer solutions for the equation $|5x^2 - y^2| = 4$
Positive integer $n$ such that $2n+1$ , $3n+1$ are both perfect squares
Maps of primitive vectors and Conway's river, has anyone built this in SAGE?
Infinitely many systems of $23$ consecutive integers
Solve the following equation for x and y:  <1,-1,-1>
Finding integers of the form $3x^2 + xy - 5y^2$ where $x$ and $y$ are integers, using diagram via arithmetic progression
Small integral representation as $x^2-2y^2$ in Pell's equation
Solving the equation $ x^2-7y^2=-3 $ over integers
Solutions to Diophantine Equations
How to prove that the roots of this equation are integers?
Does the Pell-like equation $X^2-dY^2=k$ have a simple recursion like $X^2-dY^2=1$?
http://math.stackexchange.com/questions/1737385/if-d1-is-a-squarefree-integer-show-that-x2-dy2-c-gives-some-bounds-i/1737824#1737824 "seeds"
Find all natural numbers $n$ such that $21n^2-20$ is a perfect square.
Is there a simple proof that if $(b-a)(b+a) = ab - 1$, then $a, b$ must be Fibonacci numbers? 1,1,-1; 1,11
To find all integral solutions of $3x^2 - 4y^2 = 11$
How do we solve pell-like equations?
Diophantine equation $x^2 + xy − 3y^2 = 17$ <1,1,-3>
</s>