Example 2. Let $P$ be a moving point on the parabola $y=x^{2}$, construct an equilateral triangle $O P Q (O, P, Q$ in counterclockwise order $)$, find the equation of the locus of its center.

 untranslated text:
例2. 设 $P$ 为挞物线 $y=x^{2}$ 上的动点,作正三角形 $O P Q(O 、 P 、 Q$ 按逆时针方向 $)$,求其中心的轨迹方程.

 translated text:
Example 2. Let $P$ be a moving point on the parabola $y=x^{2}$, construct an equilateral triangle $O P Q (O, P, Q$ in counterclockwise order $)$, find the equation of the locus of its center.
Let $P$ correspond to the complex number $z_{\mathrm{P}}=t+t^{2} i$, then
$$
\begin{aligned}
z_{Q} & =\left(t+t^{2} i\right)\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right) \\
= & \left(\frac{t}{2}-\frac{\sqrt{3}}{2} t^{2}\right) \\
& +\left(\sqrt{3} \frac{3}{2} t+\frac{t^{2}}{2}\right) i .
\end{aligned}
$$

If $\triangle O P \cong$ the center is $(x, y)$, then
$$
x=\frac{t}{2}-\frac{\sqrt{3}}{6} t^{2}, \quad y=\frac{\sqrt{3}}{6} t+\frac{1}{2} t^{2} \text {. }
$$

Eliminating the parameter $t$ yields
$$
\begin{aligned}
3 \sqrt{3} & x^{2}+6 x y+\sqrt{3} y^{2}+2 x \\
& -2 \sqrt{3} y=0
\end{aligned}
$$