5. The ellipse $x^{2}+2 y^{2}=3$ rotates around the $y$-axis to form a surface of revolution $\Sigma$. Let $P(x, y, z)$ be a point on $\Sigma$, then $x^{2}+2 y^{2}+z^{2}=$ $\qquad$ .
Given the ellipse intersects the $y$-axis at points $A, B$, draw $P M \perp y$-axis at point $M$, and draw $M Q / / x$-axis intersecting the ellipse at point $Q$. Then point $P$ lies on the circle with center $M$ and radius $M Q$.
Thus $x^{2}+z^{2}=|M Q|^{2}=3-2 y^{2}$ $\Rightarrow x^{2}+2 y^{2}+z^{2}=3$.