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