One class of planar rational involutions

Our aim is to find one class of planar rational involutions φλ:R2∖(L1∪L2)→R2∖(L1∪L2)φλ:R2∖(L1∪L2)→R2∖(L1∪L2) of the form φλ(x,y):=(a1x+a2y+a3a4x+a5y+a6,b1y+b2x+b3b4y+b5x+b6), where λ=(a1,…,a6,b1,…,b6)∈R12λ=(a1,…,a6,b1,…,b6)∈R12, and L1L1 and L2L2 denote two lines, a4x+a5y+a6=0a4x+a5y+a6=0 and b4y+b5x+b6=0b4y+b5x+b6=0 respectively. We give necessary and sufficient conditions on the 12 parameters ai,bi,i=1,2,…,6ai,bi,i=1,2,…,6, such that φλφλ is an involution.