On existence of primitive normal elements of rational form over finite fields of even characteristic

Let $q$ be an even prime power and $m\ge 2$ 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)$,wheretherankofthematrix$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 $\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\end{array}\right)$ if $q=2$ and $m$ is odd, and then we provide an explicit list of possible and genuine exceptional pairs $(q,m)$.