 Lozi fixed points and the alternating-sign two-cycle

The positive fixed point exists when $b<1+a$ and equals $(1,b)/(1+a-b)$; the negative one exists when $b>1-a$ and equals $(1,b)/(1-a-b)$. Equality on either boundary does not give a branch, because its fixed-point equation becomes $0=1$. For a genuine two-cycle and $a^2\ne (1-b{)}^2$, subtracting its scalar equations rules out equal signs of the two first coordinates. Solving with $x_1>0>x_2$ gives the displayed coordinates and second coordinates $y_1=b{x}_2$, $y_2=b{x}_1$. Their signs require exactly $a>|1-b|$. The cycle is unique up to interchanging its two points. Degenerate boundary parameter lines can support additional period-two families and are outside this statement.