  Given systems of two (inhomogeneous) quadratic equations in four variables,
it is known that the Hasse principle for integral points may fail. Sometimes
this failure can be explained by some integral Brauer-Manin obstruction. We
study the existence of a non-trivial algebraic part of the Brauer group for a
family of such systems and show that the failure of the integral Hasse
principle due to an algebraic Brauer-Manin obstruction is rare, as for a
generic choice of a system the algebraic part of the Brauer-group is trivial.
We use resolvent constructions to give quantitative upper bounds on the number
of exceptions.
