  Let $(M, g)$ be a Kaehler manifold whose associated Kaehler form $\omega$ is
integral and let $(L, h)\rightarrow (M, \omega)$ be a quantization hermitian
line bundle. In this paper we study those Kaehler manifolds $(M, g)$ admitting
a finite TYCZ expansion. We show that if the TYCZ expansion is finite then
$T_{mg}$ is indeed a polynomial in $m$ of degree $n$, $n=dim M$, and the
log-term of the Szeg\"{o} kernel of the disc bundle $D\subset L^*$ vanishes
(where $L^*$ is the dual bundle of $L$). Moreover, we provide a complete
classification of the Kaehler manifolds admitting finite TYCZ expansion either
when $M$ is a complex curve or when $M$ is a complex surface with a cscK metric
which admits a radial Kaehler potential.
