4. Find all integer solutions to the indeterminate equation $5 x^{2}-14 y^{2}=11 z^{2}$.
4.  Taking both sides modulo 7, we know $5 x^{2} \equiv 11 z^{2} \equiv 4 z^{2}(\bmod 7)$. If $7 \times x$, then $\left(2 z x^{-1}\right)^{2} \equiv 5(\bmod 7)$, where $x^{-1}$ is the modular inverse of $x$ modulo 7. However, a square number $\equiv 0,1$ or $4(\bmod 7)$, which is a contradiction. Therefore, $7 \mid x$. Consequently, we can deduce that $7|z, 7| y$. Thus, $\left(\frac{x}{7}, \frac{y}{7}, \frac{z}{7}\right)$ is also an integer solution to the equation. By induction, the only solution to the equation is $(x, y, z)=(0, 0,0)$.