  We consider the Brauer-Manin obstruction to the existence of integral points
on affine surfaces defined by $x^2 - ay^2 = P(t)$ over a number field. We
enumerate the possibilities for the Brauer groups of certain families of such
surfaces, and show that in contrast to their smooth compactifications, the
Brauer groups of these affine varieties need not be generated by cyclic (e.g.
quaternion) algebras. Concrete examples are given of such surfaces over
$\mathbb{Q}$ which have solutions in $\mathbb{Z}_{p}$ for all $p$ and solutions
in $\mathbb{Q}$, but for which the failure of the integral Hasse principle
cannot be explained by a Brauer-Manin obstruction. The methods of this paper
build on the ideas of several recent papers in the literature: we study the
Brauer groups (modulo constant algebras) of affine Ch\^atelet surfaces with a
focus on explicitly representing, by central simple algebras over the function
field of $X$, a finite set of classes of the Brauer group which generate the
quotient. Notably, we develop techniques for constructing explicit
representatives of non-cyclic Brauer classes on affine surfaces, as well as
provide an effective algorithm for the computation of local invariants of these
Brauer classes via effective lifting of cocycles in Galois cohomology.
