To solve the problem, we aim to determine all pairs $(m,n)$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. According to the problem statement, a set $A$ of integers is admissible if whenever $x$ and $y$ are in $A$, $x^2 + kxy + y^2$ is also in $A$ for every integer $k$.

### Step-by-Step Analysis

1. Definition of Admissible Set

Given the definition, for any integers $x, y \in A$, the expression $x^2 + kxy + y^2$ must also be in $A$ for any integer $k$. Notably, choosing specific values for $k$ yields several important cases:

- When $k = 0$, this yields $x^2 + y^2 \in A$.
- When $k = 1$, we obtain $x^2 + xy + y^2 \in A$.

2. Exploring Consequences

We compute some values to understand the closure of $A$ under these conditions:

- Starting with elements $m$ and $n$ in $A$:
- Using the condition $k = 0$, both $m^2 + n^2$ and $n^2 + m^2 = 2n^2$ must be in $A$.
- Utilizing $k = -1$, we derive:
$$m^2 - mn + n^2 \in A.$$

- If we choose $k$ such that the expression includes forms like Euclidean algorithms, this could result in generating 1 if $m$ and $n$ are coprime:

- Particularly, repeated applications will eventually include elements such as the greatest common divisor of $m$ and $n$.

3. Condition for Admissibility

The minimal condition for a set containing $m$ and $n$ to be closed under these operations is $\gcd(m, n) = 1$. This means:

- With $\gcd(m,n) = 1$, elliptic stepping continually reduces combinations of $(m, n)$ down to $\gcd(m,n)$.
- Hence, this process can eventually generate any integer, showing $A$ must be the set of all integers.

4. Conclusion

The problem therefore reduces to determining when any elements $m$ and $n$ can generate the full set of integers. This happens precisely when:

$$\gcd(m, n) = 1.$$

Thus, the set of pairs $(m, n)$ such that the only admissible set containing both $m$ and $n$ is the set of all integers is exactly those pairs for which $\gcd(m, n) = 1$. Consequently, the answer is:

$$\boxed{\text{All pairs } (m, n) \text{ of nonzero integers such that } \gcd(m, n) = 1.}$$