  In spectral theory, $j$-monotonic families of $2\times 2$ matrix functions
appear as transfer matrices of many one-dimensional operators. We present a
general theory of such families, in the perspective of canonical systems in
Arov gauge. This system resembles a continuum version of the Schur algorithm,
and allows to restore an arbitrary Schur function along the flow of associated
boundary values at infinity. In addition to results in Arov gauge, this
provides a gauge-independent perspective on the Krein-de Branges formula and
the reflectionless property of right limits on the absolutely continuous
spectrum. This work has applications to inverse spectral problems which have
better behavior with respect to a normalization at an internal point of the
resolvent domain.
