Determine the greatest number m such that the
system $x^2$ + $y^2$ = 1; |$x^3$-$y^3$|+|x-y|=$m^3$ has a solution.
To determine the greatest number \( m \) such that the system 
\[ x^2 + y^2 = 1 \]
\[ |x^3 - y^3| + |x - y| = m^3 \]
has a solution, we need to find the maximum value of \( |x^3 - y^3| + |x - y| \) under the constraint \( x^2 + y^2 = 1 \).

1. **Rewrite the expression \( |x^3 - y^3| + |x - y| \):**
   \[
   |x^3 - y^3| + |x - y| = |x - y|(x^2 + xy + y^2 + 1)
   \]
   This follows from the factorization of \( x^3 - y^3 \):
   \[
   x^3 - y^3 = (x - y)(x^2 + xy + y^2)
   \]

2. **Substitute \( x^2 + y^2 = 1 \):**
   \[
   x^2 + y^2 = 1 \implies x^2 + y^2 + 1 = 2
   \]
   Therefore, the expression becomes:
   \[
   |x - y|(2 + xy)
   \]

3. **Parameterize the variables:**
   Let \( x = \cos \theta \) and \( y = \sin \theta \) for some angle \( \theta \). This satisfies the constraint \( x^2 + y^2 = 1 \).

4. **Calculate \( |x - y| \) and \( xy \):**
   \[
   |x - y| = |\cos \theta - \sin \theta|
   \]
   \[
   xy = \cos \theta \sin \theta
   \]

5. **Substitute these into the expression:**
   \[
   |x - y|(2 + xy) = |\cos \theta - \sin \theta| \left(2 + \cos \theta \sin \theta \right)
   \]

6. **Maximize the expression:**
   To find the maximum value, we need to analyze the function:
   \[
   f(\theta) = |\cos \theta - \sin \theta| \left(2 + \cos \theta \sin \theta \right)
   \]

   Consider the critical points and the behavior of the function over the interval \( [0, 2\pi] \).

   - When \( \theta = \frac{\pi}{4} \), \( \cos \theta = \sin \theta = \frac{\sqrt{2}}{2} \):
     \[
     |\cos \theta - \sin \theta| = 0
     \]
     \[
     f\left(\frac{\pi}{4}\right) = 0 \cdot \left(2 + \frac{1}{2}\right) = 0
     \]

   - When \( \theta = 0 \) or \( \theta = \pi \), \( \cos \theta = \pm 1 \) and \( \sin \theta = 0 \):
     \[
     |\cos \theta - \sin \theta| = 1
     \]
     \[
     f(0) = 1 \cdot (2 + 0) = 2
     \]

   - When \( \theta = \frac{3\pi}{4} \) or \( \theta = \frac{5\pi}{4} \), \( \cos \theta = -\frac{\sqrt{2}}{2} \) and \( \sin \theta = \frac{\sqrt{2}}{2} \):
     \[
     |\cos \theta - \sin \theta| = \sqrt{2}
     \]
     \[
     f\left(\frac{3\pi}{4}\right) = \sqrt{2} \left(2 - \frac{1}{2}\right) = \sqrt{2} \cdot \frac{3}{2} = \frac{3\sqrt{2}}{2}
     \]

   - When \( \theta = \frac{\pi}{2} \) or \( \theta = \frac{3\pi}{2} \), \( \cos \theta = 0 \) and \( \sin \theta = \pm 1 \):
     \[
     |\cos \theta - \sin \theta| = 1
     \]
     \[
     f\left(\frac{\pi}{2}\right) = 1 \cdot (2 + 0) = 2
     \]

   The maximum value of \( f(\theta) \) is \( 2 \).

7. **Conclusion:**
   The greatest number \( m \) such that the system has a solution is:
   \[
   m^3 = 2 \implies m = \sqrt[3]{2}
   \]

The final answer is \( \boxed{ \sqrt[3]{2} } \)