Find all integer sequences of the form $ x_i, 1 \le i \le 1997$, that satisfy $ \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k$.
1. **Understanding the problem:**
   We need to find all integer sequences \( x_i \) for \( 1 \le i \le 1997 \) that satisfy the equation:
   \[
   \sum_{k=1}^{1997} 2^{k-1} x_k^{1997} = 1996 \prod_{k=1}^{1997} x_k
   \]

2. **Using the hint:**
   The hint suggests that if \((x_1, x_2, \ldots, x_{1997})\) is a solution, then \(\left(\frac{x_1}{2}, \frac{x_2}{2}, \ldots, \frac{x_{1997}}{2}\right)\) is also a solution. This implies that the sequence can be scaled by a factor of \(\frac{1}{2}\) and still satisfy the equation.

3. **Exploring the implications of the hint:**
   Let's assume that \((x_1, x_2, \ldots, x_{1997})\) is a solution. Then, substituting \(\left(\frac{x_1}{2}, \frac{x_2}{2}, \ldots, \frac{x_{1997}}{2}\right)\) into the equation, we get:
   \[
   \sum_{k=1}^{1997} 2^{k-1} \left(\frac{x_k}{2}\right)^{1997} = 1996 \prod_{k=1}^{1997} \left(\frac{x_k}{2}\right)
   \]
   Simplifying the left-hand side:
   \[
   \sum_{k=1}^{1997} 2^{k-1} \left(\frac{x_k}{2}\right)^{1997} = \sum_{k=1}^{1997} 2^{k-1} \frac{x_k^{1997}}{2^{1997}} = \sum_{k=1}^{1997} \frac{2^{k-1} x_k^{1997}}{2^{1997}}
   \]
   \[
   = \frac{1}{2^{1997}} \sum_{k=1}^{1997} 2^{k-1} x_k^{1997}
   \]
   Simplifying the right-hand side:
   \[
   1996 \prod_{k=1}^{1997} \left(\frac{x_k}{2}\right) = 1996 \frac{\prod_{k=1}^{1997} x_k}{2^{1997}}
   \]

4. **Equating both sides:**
   \[
   \frac{1}{2^{1997}} \sum_{k=1}^{1997} 2^{k-1} x_k^{1997} = 1996 \frac{\prod_{k=1}^{1997} x_k}{2^{1997}}
   \]
   Multiplying both sides by \(2^{1997}\):
   \[
   \sum_{k=1}^{1997} 2^{k-1} x_k^{1997} = 1996 \prod_{k=1}^{1997} x_k
   \]
   This is the original equation, confirming that the hint is valid.

5. **Considering integer solutions:**
   Since the sequence can be scaled by \(\frac{1}{2}\) and still be a solution, and we are looking for integer solutions, the only possibility is that all \(x_i\) must be zero. This is because any non-zero integer scaled by \(\frac{1}{2}\) would not remain an integer.

6. **Verifying the solution:**
   If \(x_i = 0\) for all \(i\), then:
   \[
   \sum_{k=1}^{1997} 2^{k-1} x_k^{1997} = \sum_{k=1}^{1997} 2^{k-1} \cdot 0 = 0
   \]
   \[
   1996 \prod_{k=1}^{1997} x_k = 1996 \cdot 0 = 0
   \]
   Both sides of the equation are equal, confirming that \(x_i = 0\) for all \(i\) is indeed a solution.

The final answer is \( \boxed{ x_i = 0 } \) for all \(i\).