This question is motivated by a remark of Bill Dubuque on my answer to the following question:
Vieta jumping is based on the idea that if you have a quadratic equation with integer coefficients that is symmetric in $x$ and $y$, you can try to find a descent argument by replacing $x$ by the other root of the equation and then changing the roles of $x$ and $y$.
To quote Bill Dubuque's comment:
"Vieta jumping" is a strange name for what is essentially descent in a group of integer points on a conic (realized by reflection). This is a special case of results on Pell equations (or equivalent theories).
And here is finally my question:
Does the Pell equation point of view give a good characterisation when this procedure will actually give a descent?