  Let $q$ be an even prime power and $m\geq2$ an integer. By $\mathbb{F}_q$, we
denote the finite field of order $q$ and by $\mathbb{F}_{q^m}$ its extension
degree $m$. In this paper we investigate the existence of a primitive normal
pair $(\alpha, \, f(\alpha))$, with $f(x)= \dfrac{ax^2+bx+c}{dx+e} \in
\mathbb{F}_{q^m}(x)$, where the rank of the matrix
  $F= \begin{pmatrix}a \, &b\, & c\\ 0\, &d \, &e \end{pmatrix}$ $\in M_{2
\times 3}(\Fm) $ is 2. Namely, we establish sufficient conditions to show that
nearly all fields of even characteristic possess such elements, except for
$\begin{pmatrix} 1 \, &1 \, & 0\\ 0\, &1 \, &0 \end{pmatrix}$ if $q=2$ and $m$
is odd, and then we provide an explicit list of possible and genuine
exceptional pairs $(q,m)$.
