In his 1854 paper, "Deuxième mémoire sur les fonctions doublement périodiques" ("Second memoir on doubly periodic functions"), Cayley discusses (what we would today describe as) a certain class of transformations $\mathbb C^2 \rightarrow \mathbb C^2$ corresponding to matrices $$ \begin{bmatrix} \lambda & \mu \\ \nu & \rho \\ \end{bmatrix} $$ with integer entries. Up to sign, Cayley requires the determinant $\lambda\rho-\mu\nu$ of such a matrix to be an odd prime number. Moreover, for every pair of complex numbers $\left(\Omega,\Upsilon\right)$, all such transformations are required to satisfy the conditions $$\lambda\Omega+\mu\Upsilon=\left(2k+1\right)\Omega,$$ $$\nu\Omega+\rho\Upsilon=\left(2k+1\right)\Upsilon,$$ where $2k+1=\left|\lambda\rho-\mu\nu\right|$ is a positive odd prime, equal up to sign to the matrix determinant.
Cayley calls such a transformation ‘proper’ (propre) if $2k+1=\lambda\rho-\mu\nu$ (i.e., if the determinant is itself positive), and ‘regular’ (régulière) if both $\lambda$ and $\rho$ are odd and both $\mu$ and $\nu$ are even. He says that all transformations mentioned in the "Deuxième Mémoire" will be assumed proper and regular. After introducing these conditions and making some basic observations, Cayley says, “Je suppose d'abord que $2k+1$ soit égal à l'unité, transformation que l’on peut nommer triviale.” (“I suppose first that $2k+1$ may be equal to unity, a transformation that one could call trivial.”)
In other words, a trivial transformation in Cayley's sense is a proper and regular map $\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ corresponding to an integer matrix with determinant $1$, and such that the matrix entries satisfy the two equations mentioned above.
My question (finally) is this: Besides the transformation corresponding to the identity matrix $$\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right],$$ what are some examples of trivial transformations in Cayley's sense? What do such transformations look like in general?