  Let $\Sigma=(X,\mathcal B)$ a $4$-cycle system of order $v=1+8k$. A
$c$-colouring of type $s$ is a map $\phi\colon \mathcal B\rightarrow \mathcal
C$, with $C$ set of colours, such that exactly $c$ colours are used and for
every vertex $x$ all the blocks containing $x$ are coloured exactly with $s$
colours. Let $4k=qs+r$, with $q,r\ge 0$. $\phi$ is \emph{equitable} if for
every vertex $x$ the set of the $4k$ blocks containing $x$ is parted in $r$
colour classes of cardinality $q+1$ and $s-r$ colour classes of cardinality
$q$. In this paper we study colourings for which $s|k$, giving a description of
equitable block colourings for $c\in
\{s,s+1,\dots,\lfloor\tfrac{2s^2+s}{3}\rfloor \}$.
