  The relation between the Toda lattices and similar nonlinear chains and
orthogonal polynomials on the real line has been elaborated immensely for the
last decades. We examine another system of the differential-difference
equations known as the Schur flow within the framework of the theory of
orthogonal polynomials on the unit circle. This system can be exhibited in
equivalent form as the Lax equation, and the corresponding spectral measure
undergoes a simple transformation. The general result is illustrated on the
modified Bessel measures on the unit circle and the long time behavior of their
Verblunsky coefficients.
