  It is shown that, under some natural additional conditions, an operator which
intertwines one cyclic singular unitary operator with one dimensional
perturbation of another cyclic singular unitary operator is the operator of
multiplication by a multiplier between model spaces. Using this result, it is
shown that if $T$ is one dimensional perturbation of a unitary operator, $T$ is
a quasiaffine transform of a singular unitary operator, and $T$ is power
bounded, then $T$ is similar to a unitary operator, and $\sup_{n\geq
0}\|T^{-n}\|\leq(2(\sup_{n\geq 0}\|T^n\|)^2+1)\cdot(\sup_{n\geq 0}\|T^n\|)^5$.
