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We expand the class of curves $(\varphi_1(t),\varphi_2(t)),\ t\in[0,1]$ forwhich the $\ell^2$ decoupling conjecture holds for $2\leq p\leq 6$. Our classof curves includes all real-analytic regular curves with isolated points ofvanishing curvature and all curves of the form $(t,t^{1+\nu})$ for $\nu\in(0,\infty)$.

$\ell^2$ Decoupling in $\mathbb R^2$ for curves with Vanishing Curvature