Given a binary serie $A=a_1a_2...a_k$ is called "symmetry" if $a_i=a_{k+1-i}$ for all $i=1,2,3,...,k$, and $k$ is the length of that binary serie. If $A=11...1$ or $A=00...0$ then it is called "special". Find all positive integers $m$ and $n$ such that there exist non "special" binary series $A$ (length $m$) and $B$ (length $n$) satisfying when we place them next to each other, we receive a "symmetry" binary serie $AB$
To solve this problem, we need to find all positive integers \( m \) and \( n \) such that there exist non-"special" binary series \( A \) (of length \( m \)) and \( B \) (of length \( n \)) which, when concatenated, form a "symmetry" binary series \( AB \).

1. **Understanding the Symmetry Condition**:
   - A binary series \( A = a_1a_2 \ldots a_k \) is called "symmetry" if \( a_i = a_{k+1-i} \) for all \( i = 1, 2, \ldots, k \).
   - \( A \) is "special" if it consists entirely of 1s or entirely of 0s.

2. **Concatenation and Symmetry**:
   - Let \( A = a_1a_2 \ldots a_m \) and \( B = b_1b_2 \ldots b_n \).
   - The concatenated series \( AB = a_1a_2 \ldots a_m b_1b_2 \ldots b_n \) must be symmetric.
   - This implies \( a_i = b_{n+1-i} \) for \( i = 1, 2, \ldots, m \) and \( b_i = a_{m+1-i} \) for \( i = 1, 2, \ldots, n \).

3. **Parity Consideration**:
   - If \( m \) and \( n \) have the same parity, then \( m+n \) is even.
   - The middle position of \( AB \) is \( \frac{m+n}{2} \), which must be symmetric.

4. **Constructing Non-Special Series**:
   - If \( m \) and \( n \) are both odd, then \( \frac{m+n}{2} \) is an integer.
   - We can place \( 01 \) or \( 10 \) at the middle and expand symmetrically.
   - For example, if \( m = 3 \) and \( n = 3 \), we can have \( A = 010 \) and \( B = 010 \), forming \( AB = 010010 \), which is symmetric and non-special.

5. **General Case**:
   - For any odd \( m \) and \( n \), we can construct non-special symmetric series by ensuring the middle part is \( 01 \) or \( 10 \).

6. **Conclusion**:
   - The solution works for any odd \( m \) and \( n \).

The final answer is \( \boxed{ m } \) and \( n \) are both odd positive integers.