Determine the integers $n\geqslant 2$ for which the equation $x^2-\hat{3}\cdot x+\hat{5}=\hat{0}$ has a unique solution in $(\mathbb{Z}_n,+,\cdot).$
To determine the integers \( n \geq 2 \) for which the equation \( x^2 - \hat{3} \cdot x + \hat{5} = \hat{0} \) has a unique solution in \( (\mathbb{Z}_n, +, \cdot) \), we proceed as follows:

1. **Identify the condition for unique solutions:**
   If \( a \) is a solution, then \( \hat{3} - a \) is also a solution. For the solution to be unique, we must have \( a = \hat{3} - a \). This implies:
   \[
   2a = \hat{3}
   \]
   Therefore, \( a = \frac{\hat{3}}{2} \) must be a well-defined element in \( \mathbb{Z}_n \).

2. **Check the consistency of the equation:**
   Substitute \( a = \frac{\hat{3}}{2} \) into the original equation:
   \[
   x^2 - \hat{3} \cdot x + \hat{5} = \hat{0}
   \]
   We multiply the entire equation by 4 to clear the fraction:
   \[
   4(x^2 - \hat{3} \cdot x + \hat{5}) = \hat{0}
   \]
   Simplifying, we get:
   \[
   4x^2 - 4 \cdot \hat{3} \cdot x + 4 \cdot \hat{5} = \hat{0}
   \]
   \[
   4x^2 - \hat{12}x + \hat{20} = \hat{0}
   \]

3. **Simplify the equation:**
   We know that \( 4x^2 - \hat{12}x + \hat{20} = \hat{0} \) must hold in \( \mathbb{Z}_n \). This simplifies to:
   \[
   \hat{9} - \hat{18} + \hat{20} = \hat{0}
   \]
   \[
   \hat{11} = \hat{0}
   \]
   This implies that \( n \mid 11 \).

4. **Determine the possible values of \( n \):**
   Since \( n \mid 11 \) and \( n \geq 2 \), the only possible value for \( n \) is:
   \[
   n = 11
   \]

5. **Verify the solution for \( n = 11 \):**
   For \( n = 11 \), the equation becomes:
   \[
   x^2 - \hat{3}x + \hat{5} = \hat{0}
   \]
   In \( \mathbb{Z}_{11} \), this is equivalent to:
   \[
   x^2 + \hat{8}x + \hat{16} = \hat{0}
   \]
   This can be rewritten as:
   \[
   (x + \hat{4})^2 = \hat{0}
   \]
   Clearly, this equation has a unique solution \( x = \hat{7} \) in \( \mathbb{Z}_{11} \).

The final answer is \( \boxed{ n = 11 } \).