Linear spaces with a 2-line: use the grid model through its two points

To decide whether a small linear space exists that contains a line of size 2 = {u, v}, label every other point x by the pair (line ux, line vx). Two distinct points cannot share both lines, so the points become cells of a grid whose rows are u's lines and whose columns are v's lines; row and column sizes are the line sizes minus one. Every remaining line is a transversal (no two cells in a row or column), and these lines must cover each transversal pair exactly once. The local equation also forces each cell's remaining line counts from its (row size, column size) type. A line meets each row at most once, which bounds the per-row sums. The model needs no clever symmetry breaking: enumerate 0/1 matrices with the given margins (rows sorted within a size class), then run exact cover. For 14 points it found 148 grids and settled everything in seconds, and it independently confirmed a double-lex incidence search. Running the large lines alone first shows which cases are already killed by them.