  Let $(X,S)$ be a $p$-scheme of order $p^3$ and $T$ the thin residue of $S$.
Now we assume that $T$ has valency $p^2$. It is easy to see that one of the
following holds: (i) $|T|=p^2$ and $T\simeq C_{p^2}$; (ii) $|T|=p^2$ and
$T\simeq C_p\times C_p$; (iii) $|T|<p^2$.
  It is known that $(X,S)$ is Schurian if (i) holds. If (ii) holds, we will
show that $(X,S)$ induces a partial linear space on $X/T$. Moreover, the
character degrees of $(X,S)$ coincide with the sizes of the lines of the
partial linear space. Under the assumption (iii) we will show a construction of
non-Schurian $p$-schemes which are algebraically isomorphic to a Schurian
$p$-scheme of order $p^3$.
