Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and
\[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots
\]

Determine real number $ a$ such that if $ x_1 > a$, then the sequence is monotonically decreasing, and if $ 0 < x_1 < a$, then the sequence is not monotonic.
1. Given the sequence \( (x_1, x_2, \cdots) \) of positive numbers such that \( (8x_2 - 7x_1)x_1^7 = 8 \) and 
   \[
   x_{k+1}x_{k-1} - x_k^2 = \frac{x_{k-1}^8 - x_k^8}{x_k^7 x_{k-1}^7} \text{ for } k = 2, 3, \ldots
   \]
   we need to determine a real number \( a \) such that if \( x_1 > a \), then the sequence is monotonically decreasing, and if \( 0 < x_1 < a \), then the sequence is not monotonic.

2. First, we simplify the given recurrence relation:
   \[
   x_{k+1}x_{k-1} - x_k^2 = \frac{x_{k-1}^8 - x_k^8}{x_k^7 x_{k-1}^7}
   \]
   Multiplying both sides by \( x_k^7 x_{k-1}^7 \), we get:
   \[
   x_{k+1} x_{k-1} x_k^7 x_{k-1}^7 - x_k^9 x_{k-1}^7 = x_{k-1}^8 - x_k^8
   \]
   Simplifying further:
   \[
   x_{k+1} x_k^7 x_{k-1}^8 - x_k^9 x_{k-1}^7 = x_{k-1}^8 - x_k^8
   \]
   Rearranging terms:
   \[
   x_{k+1} x_k^7 x_{k-1}^8 = x_k^9 x_{k-1}^7 + x_{k-1}^8 - x_k^8
   \]
   Dividing both sides by \( x_{k-1}^8 \):
   \[
   x_{k+1} x_k^7 = x_k^9 \frac{x_{k-1}^7}{x_{k-1}^8} + 1 - \frac{x_k^8}{x_{k-1}^8}
   \]
   Simplifying:
   \[
   x_{k+1} x_k^7 = x_k^9 \frac{1}{x_{k-1}} + 1 - \left(\frac{x_k}{x_{k-1}}\right)^8
   \]
   \[
   x_{k+1} x_k^7 = x_k^8 \left(\frac{x_k}{x_{k-1}} - \left(\frac{x_k}{x_{k-1}}\right)^8\right) + 1
   \]

3. Given the initial condition \( (8x_2 - 7x_1)x_1^7 = 8 \), we can solve for \( x_2 \):
   \[
   8x_2 - 7x_1 = \frac{8}{x_1^7}
   \]
   \[
   x_2 = \frac{8}{8} \cdot \frac{1}{x_1^7} + \frac{7}{8} x_1
   \]
   \[
   x_2 = \frac{1}{x_1^7} + \frac{7}{8} x_1
   \]

4. We now generalize this to find \( x_{k+1} \):
   \[
   x_{k+1} = \frac{1}{x_k^7} + \frac{7}{8} x_k
   \]

5. Define the function \( f(x) = \frac{7}{8}x + \frac{1}{x^7} \). We need to analyze the behavior of this function:
   - If \( x = 8^{1/8} \), then \( f(x) = x \).
   - If \( x > 8^{1/8} \), then \( f(x) < x \), indicating a decreasing sequence.
   - If \( x < 8^{1/8} \), then \( f(x) > x \), indicating an increasing sequence initially, but it will eventually decrease.

6. Therefore, the critical value \( a \) is \( 8^{1/8} \).

The final answer is \( \boxed{ a = 8^{1/8} } \)