Starting with the Iwasawa-type decomposition of a first-order optical system (or ABCD system) as a cascade of a lens, a magnifier, and an orthosymplectic system (a system that is both symplectic and orthogonal), a further decomposition of the orthosymplectic system in the form of a separable fractional Fourier transformer embedded between two spatial-coordinate rotators is proposed. The resulting decomposition of the entire first-order optical system then shows a physically attractive representation of the linear canonical integral transformation, which, in contrast to Collins integral, is valid for any ray transformation matrix. Any lossless first-order optical system can be described by its ray transformation matrix, 1 which relates the position r i and direction q i of an incoming ray to the position r o and direction q o of the outgoing ray:The ray transformation matrix T of such a system is real and symplectic, yielding the relationsUsing the submatrices A, B, and D, and assuming that B is a nonsingular matrix, we can represent the first-order optical system by the Collins integralin which the output amplitude f o ͑r͒ is expressed in terms of the input amplitude f i ͑r͒. The phase factor exp͑i͒ in Eq. (3) has been included for completeness to cope with the optical path length and the metaplectic sign problem; unless absolutely necessary, it will usually be omitted. In the case that B equals the null matrix, B = 0, Collins integral (3) reduces toThe singular case det B = 0 with B 0, however, is rather difficult to handle. To treat the singular case, Moshinsky and Quesne 3 showed that any symplectic ABCD matrix with a singular submatrix B can be decomposed asin which BЈ is a nonsingular diagonal matrix and det͑B − BЈD͒ 0. After doing so, they could then use the Collins integral (3) for each of the two subsystems in the cascade (5) separately, thus avoiding the singular case. The way to find the diagonal matrix BЈ, however, is not easy. In this Letter we restrict ourselves to the case that A, B, C, and D are 2 ϫ 2 matrices and we propose a representation of the linear canonical integral transformation in an alternative form that is valid for any ray transformation matrix, whether or not it has a singular submatrix B. Our method is based on the Iwasawa decomposition, 4,5 followed by a further decomposition of an orthosymplectic system into a separable fractional Fourier transformer embedded between two spatial-coordinate rotators.After properly normalizing to dimensionless variables, any symplectic matrix can be decomposed in the Iwasawa form (Ref. 5, Secs. 9.5 and 10.2) in which the first matrix represents a lens described by the symmetric matrixthe second matrix represents a magnifier described by the symmetric matrixand the third matrix represents a so-called orthosymplectic system-a system that is both orthogonal and symplectic-described by the unitary matrix U = X + iY = ͑AA t + BB t ͒ −1/2 ͑A + iB͒. ͑9͒ B = SY, and since S is nonsingular, a singularity of B is only due to the orthosymplectic system.