Given a general quadratic diophantine equation of the form $$ax^2+bxy+cy^2+dx+ey+f=0$$ I am interested in the sufficient conditions for the existence of a recurrence relation of the form $$X_{n+1} = PX_n+QY_n + K$$ $$ Y_{n+1} = RX_n + SY_n +L$$ that can generate all positive integer solutions $(X_n, Y_n)$ starting from some fundamental solution $(X_0, Y_0)$. I have seen the proof that such a recurrence relation exists in the case of a Pell equation $x^2-dy^2=1$ when $d$ is not a perfect square, but for the more general case I have not seen such a proof, but rather an assumption that the recurrence relation exists.
My understanding is that a necessary set of conditions for the existence of such a recurrence relation is $a)$ that the discriminant $b^2-4ac>0$ and $b)$ that there are infinitely many positive integer solutions. My specific questions are:
$0)$ Are conditions $a)$ and $b)$ indeed necessary? $b)$ obviously is, but what about $a)$?
$1)$ Are the two conditions $a)$ and $b)$ sufficient for the existence of a recurrence relation as above? If so, could someone point me towards a proof? If not, could someone provide me with a counter-example?
$2)$ Whether this set of conditions is sufficient or not, is there some "nicer" sufficient set of conditions for the existence of such a recurrence relation? In particular, $(i)$ conditions expressed only in terms of a relation between the coefficients $a, b, c, d, e, f$ of the equation, or alternatively $(ii)$ a weaker version of $b)$, for instance that there exists at least one positive integer solution.
I am specifically interested in the sufficient conditions for the existence of the recurrence relation, and not for instance in how to find the fundamental solution $(X_0, Y_0)$, nor how to compute the coefficients $P, Q, K, R, S, L$.
