  In this paper, the $L^2$ boundedness of the Hilbert transform along variable
flat curve $(t,P(x_1)\gamma(t))$
$$H_{P,\gamma}f(x_1,x_2):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x_1-t,x_2-P(x_1)\gamma(t))\,\frac{\textrm{d}t}{t},\quad
\forall\, (x_1,x_2)\in\mathbb{R}^2,$$ is studied, where $P$ is a real
polynomial on $\mathbb{R}$. A new sufficient condition on the curve $\gamma$ is
introduced.
