  We study a family of polynomials introduced by Daigle and Freudenburg, which
contains the famous V\'en\'ereau polynomials and defines
$\mathbb{A}^2$-fibrations over $\mathbb{A}^2$. According to the
Dolgachev-Weisfeiler conjecture, every such fibration should have the structure
of a locally trivial $\mathbb{A}^2$-bundle over $\mathbb{A}^2$. We follow an
idea of Kaliman and Zaidenberg to show that these fibrations are locally
trivial $\mathbb{A}^2$-bundles over the punctured plane, all of the same
specific form $X_f$, depending on an element $f\in k[a^{\pm 1},b^{\pm 1}][x]$.
We then introduce the notion of bivariables and show that the set of
bivariables is in bijection with the set of locally trivial bundles $X_f$ that
are trivial. This allows us to give another proof of Lewis's result stating
that the second V\'en\'ereau polynomial is a variable and also to trivialise
other elements of the family $X_f$. We hope that the terminology and methods
developed here may lead to future study of the whole family $X_f$.
