Find the complex numbers $x,y,z$,with $\mid x\mid=\mid y\mid=\mid z\mid$,knowing that

$x+y+z$ and $x^{3}+y^{3}+z^{3}$ are be real numbers.
1. Let \( x = r(\cos a + i \sin a) \), \( y = r(\cos b + i \sin b) \), and \( z = r(\cos c + i \sin c) \). Given that \( |x| = |y| = |z| \), we have \( |x| = r \), \( |y| = r \), and \( |z| = r \).

2. Since \( x + y + z \) is a real number, the imaginary part of \( x + y + z \) must be zero. Therefore, we have:
   \[
   \sin a + \sin b + \sin c = 0
   \]

3. Given that \( x^3 + y^3 + z^3 \) is also a real number, the imaginary part of \( x^3 + y^3 + z^3 \) must be zero. Using the triple angle formula for sine, we have:
   \[
   x^3 = r^3 (\cos 3a + i \sin 3a)
   \]
   \[
   y^3 = r^3 (\cos 3b + i \sin 3b)
   \]
   \[
   z^3 = r^3 (\cos 3c + i \sin 3c)
   \]
   Therefore, the imaginary part of \( x^3 + y^3 + z^3 \) is:
   \[
   \sin 3a + \sin 3b + \sin 3c = 0
   \]

4. Using the triple angle identity for sine, we have:
   \[
   \sin 3a = 3 \sin a - 4 \sin^3 a
   \]
   \[
   \sin 3b = 3 \sin b - 4 \sin^3 b
   \]
   \[
   \sin 3c = 3 \sin c - 4 \sin^3 c
   \]
   Substituting these into the equation, we get:
   \[
   3 (\sin a + \sin b + \sin c) - 4 (\sin^3 a + \sin^3 b + \sin^3 c) = 0
   \]
   Since \( \sin a + \sin b + \sin c = 0 \), we have:
   \[
   - 4 (\sin^3 a + \sin^3 b + \sin^3 c) = 0
   \]
   Therefore:
   \[
   \sin^3 a + \sin^3 b + \sin^3 c = 0
   \]

5. The equation \( \sin^3 a + \sin^3 b + \sin^3 c = 0 \) implies that at least one of \( \sin a \), \( \sin b \), or \( \sin c \) must be zero. Without loss of generality, assume \( \sin a = 0 \). Then \( a = k\pi \) for some integer \( k \), and \( x \) is real.

6. If \( \sin a = 0 \), then \( \sin b + \sin c = 0 \). This implies \( \sin b = -\sin c \). Therefore, \( y \) and \( z \) are complex conjugates of each other.

7. Hence, the possible solutions are:
   - \( x, y, z \) are all real numbers.
   - \( x \) is real, and \( y \) and \( z \) are complex conjugates.

The final answer is \( \boxed{ x, y, z } \) are either all real numbers or \( x \) is real and \( y, z \) are complex conjugates.