Given the ellipse $C : x ^ { 2 } + 3 y ^ { 2 } = 3$, a line passing through point $D ( 1,0 )$ but not through point $E ( 2,1 )$ intersects the ellipse $C$ at points $A$ and $B$. The line $AE$ intersects the line $x = 3$ at point $M$.\n(1) Find the eccentricity of the ellipse $C$;\n(II) If $AB$ is perpendicular to the $x$-axis, find the slope of line $BM$;\n(III) Determine the positional relationship between line $BM$ and line $DE$, and explain the reason.