Quadratic integral solutions to double Pell equations

We study the quadratic integral points - that is, (S-)integral points defined over any extension of degree two of the base field - on a curve defined in P_3 by a system of two Pell equations. Such points belong to three families explicitly described, or belong to a finite set whose cardinality may be explicitly bounded in terms of the base field, the equations defining the curve and the set S. We exploit the peculiar geometry of the curve to adapt the proof of a theorem of Vojta, which in this case does not apply.