  In this paper, we introduce a novel non-linear uniform subdivision scheme for
the generation of curves in $\mathbb{R}^n$, $n\geq2$. This scheme is
distinguished by its capacity to reproduce second-degree polynomial data on
non-uniform grids without necessitating prior knowledge of the grid
specificities. Our approach exploits the potential of annihilation operators to
infer the underlying grid, thereby obviating the need for end-users to specify
such information. We define the scheme in a non-stationary manner, ensuring
that it progressively approaches a classical linear scheme as the iteration
number increases, all while preserving its polynomial reproduction capability.
  The convergence is established through two distinct theoretical methods.
Firstly, we propose a new class of schemes, including ours, for which we
establish $\mathcal{C}^1$ convergence by combining results from the analysis of
quasilinear schemes and asymptotically equivalent linear non-uniform
non-stationary schemes. Secondly, we adapt conventional analytical tools for
non-linear schemes to the non-stationary case, allowing us to again conclude
the convergence of the proposed class of schemes.
  We show its practical usefulness through numerical examples, showing that the
generated curves are curvature continuous.
