On the number of ${L}_{\infty\omega_1}$-equivalent non-isomorphic models

We prove that if $\operatorname{ZF}$ is consistent then $\operatorname{ZFC} + \operatorname{GCH}$ is consistent with the following statement: There is for every $k < \omega$ a model of cardinality $\aleph_1$ which is $L_{\infty{\omega_{1}}}$-equivalent to exactly $k$non-isomorphic models of cardinality $\aleph_1$. In order to get this result we introduce ladder systems and colourings different from the ``standard'' counterparts, and prove the following purely combinatorial result: For each prime number $p$ and positive integer $m$ it is consistent with $\operatorname{ZFC} + \operatorname{GHC}$ that there is a ``good'' ladder system having exactly $p^m$ pairwise nonequivalent colourings.