  Given positive integers $q,n,m$ and $a\in\mathbb{F}_{q}$, where $q$ is an odd
prime power and $n\geq 5$, we investigate the existence of a primitive normal
pair $(\epsilon,f(\epsilon))$ in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$
such that $\mathrm{STr}_{q^n/q}(\epsilon)=a$, where
$f(x)=\frac{f_{1}(x)}{f_{2}(x)}\in\mathbb{F}_{q^n}(x)$ is a rational function
together with deg$(f_{1})+$deg$(f_{2})=m$ and $\mathrm{STr}_{q^n/q}(\epsilon) =
\sum_{0\leq i<j\leq n-1}^{}\epsilon^{q^i+q^j}$. Finally, we conclude that for
$m=3$, $n\geq 6$ and $q=7^k$; $k\in\mathbb{N}$, such a pair will exist
certainly for all $(q,n)$ except at most $14$ choices.
