Determine all three integers $(x, y, z)$ that are solutions of the system 
$$x + y -z = 6$$
$$x^3 + y^3 -z^3 = 414$$
1. We start with the given system of equations:
   \[
   \begin{cases}
   x + y - z = 6 \\
   x^3 + y^3 - z^3 = 414
   \end{cases}
   \]

2. We use the identity for the sum of cubes:
   \[
   x^3 + y^3 - z^3 = (x + y - z)(x^2 + y^2 + z^2 - xy - yz - zx)
   \]
   Given \(x + y - z = 6\), we substitute this into the identity:
   \[
   x^3 + y^3 - z^3 = 6(x^2 + y^2 + z^2 - xy - yz - zx)
   \]

3. Substitute \(x^3 + y^3 - z^3 = 414\) into the equation:
   \[
   414 = 6(x^2 + y^2 + z^2 - xy - yz - zx)
   \]
   Divide both sides by 6:
   \[
   69 = x^2 + y^2 + z^2 - xy - yz - zx
   \]

4. Substitute \(z = x + y - 6\) into the equation:
   \[
   69 = x^2 + y^2 + (x + y - 6)^2 - xy - y(x + y - 6) - x(x + y - 6)
   \]

5. Simplify the equation:
   \[
   69 = x^2 + y^2 + (x^2 + y^2 + 36 + 2xy - 12x - 12y) - xy - yx - y^2 + 6y - x^2 - xy + 6x
   \]
   Combine like terms:
   \[
   69 = 2x^2 + 2y^2 + 2xy - 12x - 12y + 36 - xy - xy - y^2 + 6y - x^2 - xy + 6x
   \]
   \[
   69 = x^2 + y^2 + xy - 6x - 6y + 36
   \]
   \[
   33 = x^2 + y^2 + xy - 6x - 6y
   \]

6. We need to find integer solutions \((x, y, z)\) that satisfy both equations. We can test possible values for \(x\) and \(y\) and check if they satisfy the equations.

7. Testing possible values:
   - For \((x, y, z) = (3, 8, 5)\):
     \[
     x + y - z = 3 + 8 - 5 = 6 \quad \text{(satisfies the first equation)}
     \]
     \[
     x^3 + y^3 - z^3 = 3^3 + 8^3 - 5^3 = 27 + 512 - 125 = 414 \quad \text{(satisfies the second equation)}
     \]
   - For \((x, y, z) = (8, 3, 5)\):
     \[
     x + y - z = 8 + 3 - 5 = 6 \quad \text{(satisfies the first equation)}
     \]
     \[
     x^3 + y^3 - z^3 = 8^3 + 3^3 - 5^3 = 512 + 27 - 125 = 414 \quad \text{(satisfies the second equation)}
     \]
   - For \((x, y, z) = (3, -5, -8)\):
     \[
     x + y - z = 3 - 5 - (-8) = 6 \quad \text{(satisfies the first equation)}
     \]
     \[
     x^3 + y^3 - z^3 = 3^3 + (-5)^3 - (-8)^3 = 27 - 125 - (-512) = 414 \quad \text{(satisfies the second equation)}
     \]
   - For \((x, y, z) = (-5, 8, -3)\):
     \[
     x + y - z = -5 + 8 - (-3) = 6 \quad \text{(satisfies the first equation)}
     \]
     \[
     x^3 + y^3 - z^3 = (-5)^3 + 8^3 - (-3)^3 = -125 + 512 - (-27) = 414 \quad \text{(satisfies the second equation)}
     \]
   - For \((x, y, z) = (-5, 3, -8)\):
     \[
     x + y - z = -5 + 3 - (-8) = 6 \quad \text{(satisfies the first equation)}
     \]
     \[
     x^3 + y^3 - z^3 = (-5)^3 + 3^3 - (-8)^3 = -125 + 27 - (-512) = 414 \quad \text{(satisfies the second equation)}
     \]
   - For \((x, y, z) = (8, -5, -3)\):
     \[
     x + y - z = 8 - 5 - (-3) = 6 \quad \text{(satisfies the first equation)}
     \]
     \[
     x^3 + y^3 - z^3 = 8^3 + (-5)^3 - (-3)^3 = 512 - 125 - (-27) = 414 \quad \text{(satisfies the second equation)}
     \]

8. Therefore, the solutions are:
   \[
   (x, y, z) = (3, 8, 5), (8, 3, 5), (3, -5, -8), (-5, 8, -3), (-5, 3, -8), (8, -5, -3)
   \]

The final answer is \( \boxed{ (3, 8, 5), (8, 3, 5), (3, -5, -8), (-5, 8, -3), (-5, 3, -8), (8, -5, -3) } \)