  In a previous paper, the author gave a combinatorial proof and refinement of
Siladi\'c's theorem, a Rogers-Ramanujan type partition identity arising from
the study of Lie algebras. Here we use the basic idea of the method of weighted
words introduced by Alladi and Gordon to give a non-dilated version, further
refinement and companion of Siladi\'c's theorem. However, while in the work of
Alladi and Gordon, identities were proved by doing transformations on
generating functions, we use recurrences and $q$-difference equations as the
original method seems difficult to apply in our case. As the non-dilated
version features the same infinite product as Schur's theorem, another dilation
allows us to find a new interesting companion of Schur's theorem, with
difference conditions very different from the original ones.
