  This article investigates the existence of an $r$-primitive $k$-normal
polynomial, defined as the minimal polynomial of an $r$-primitive $k$-normal
element in $\mathbb{F}_{q^n}$, with a specified degree $n$ and two given
coefficients over the finite field $\mathbb{F}_{q}$. Here, $q$ represents an
odd prime power, and $n$ is an integer. The article establishes a sufficient
condition to ensure the existence of such a polynomial. Using this condition,
it is demonstrated that a $2$-primitive $2$-normal polynomial of degree $n$
always exists over $\mathbb{F}_{q}$ when both $q\geq 11$ and $n\geq 15$.
However, for the range $10\leq n\leq 14$, uncertainty remains regarding the
existence of such a polynomial for $71$ specific pairs of $(q,n)$. Moreover,
when $q<11$, the number of uncertain pairs reduces to $16$. Furthermore, for
the case of $n=9$, extensive computational power is employed using SageMath
software, and it is found that the count of such uncertain pairs is reduced to
$3988$.
