Solve[(1 - x^2) (-W - 3 x + Sqrt[6] Sqrt[z]) == -W^4 x (-1 + x^2) y + 3 W y (W^3 (-1 + x^2) (-1 + Sqrt[1 - x^2] y) -
  6 Sqrt[6] (y + Sqrt[1 - x^2] (-1 + z))^2 Sqrt[z] +
  W^3 (-1 + x^2) z) == 9 Sqrt[6] (y + Sqrt[1 - x^2] (-1 + z))^2 (1 - 2 z) Sqrt[z] -1/3 W^3 (-1 + x^2) z (Sqrt[6] W x - 9 (-1 + Sqrt[1 - x^2] y + z)) ==3 W (W^3 (-1 + x^2) (-1 + Sqrt[1 - x^2] y) -
 6 Sqrt[6] (y + Sqrt[1 - x^2] (-1 + z))^2 Sqrt[z] +
 W^3 (-1 + x^2) z) == 0, {x, y, z, W}]