  In this paper we study a local structure of extrmal contractions $f\colon
X\to S$ from threffolds $X$ with only terminal singularities onto a surface
$S$. If the surface $S$ is non-singular and $X$ has a unique non-Gorenstein
point on a fiber we prove that either the linear system $|-K_X|$, $|-2K_X|$ or
$|-3K_X|$ contains a "good" divisor.
