We prove that for any open Riemann surface $N$ and finite subset $Z\subset \mathbb{ S}^1=\{ z\in\mathbb{ C}\,|\;|z|=1\},$ there exist an infinite closed set $Z_N \subset \mathbb{ S}^1$ containing $Z$ and a null holomorphic curve $F=(F_j)_{ j=1,2,3}:N\to\mathbb{ C}^3$ such that the map $Y:Z_N\times N\to \mathbb{ R}^2,$ $Y(v,P)=Re(v(F_1,F_2)(P)),$ is proper. In particular, $Re(vF):N \to\mathbb{ R}^3$ is a proper conformal minimal immersion properly projecting into $\mathbb{ R}^2=\mathbb{ R}^2\times\{ 0\}\subset\mathbb{ R}^3,$ for all $v \in Z_N.$
Categories: math.DG