Existence of primitive normal pairs over finite fields with prescribed subtrace

Given positive integers q, n, m and aₐ, where q is an odd prime power and n 5, we investigate the existence of a primitive normal pair (, f () ) in Fₐ^₍ over Fₐ such that STrₐ䂞/ₐ () =a, where f (x) =f₁ (ₗ) f₂ (x) ₐ䂞 (x) is a rational function together with deg (f₁) +deg (f₂) =m and STrₐ䂞/ₐ () = ₀ ₈<₉ ₍-₁^^qⁱ+qʲ. Finally, we conclude that for m=3, n 6 and q=7ᵏ; k, such a pair will exist certainly for all (q, n) except at most 14 choices.