Mirror property for nonsingular mixed configurations of lines and points in \(\mathbb{R}^ 3\) (Q1327454)

Let \(M(h,k)\) denote the topological space whose points are the sets of \(h\) lines and \(k\) points of \(\mathbb{R}^ 3\) in general position. An element of \(M(h,k)\) is said to be mirror if there exists a continuous path in \(\mathbb{R}^ 3\) joining it with its image after reflection in a plane of \(\mathbb{R}^ 3\), and nonmirror otherwise. The question considered here is which \(M(h,k)\) contains mirror elements and which ones contain nonmirror elements. The problem is solved for all cases except when \(h\equiv 1\pmod 4\) with \(h\geq 5\) and \(k=2\) or 3.