  This paper aims to investigate the evolution problem for planar curves with
singularities. Motivated by the inverse curvature flow introduced by Li and
Wang (Calc. Var. Partial Differ. Equ. 62 (2023), No. 135), we intend to
consider the area-preserving and length-preserving inverse curvature flow with
nonlocal term for $\ell$-convex Legendre curves. For the area-preserving flow,
an $\ell$-convex Legendre curve %of with initial algebraic area $A_0>0$ evolves
to a circle of radius $\sqrt{\frac{A_0}{\pi}}$. For the length-preserving flow,
an $\ell$-convex Legendre curve %of with initial algebraic length $L_0$ evolves
to a circle of radius $\frac{L_0}{2\pi}$. As the by-product, we obtain some
geometric inequalities for $\ell$-convex Legendre curves through the
length-preserving flow.
