Summary: The Novikov problem for a superintuitionistic logic \(L\) is to describe the class of all maximal conservative (i.e. P. S. Novikov complete) extensions of \(L\) in the language with additional logical connectives and logical constants. Since the family of all superintuitionistic logics has the power of the continuum, it is sensible to apply the P. S. Novikov problem to superintuitionistic logics which for one reason or other have already come to researchers/ attention. In particular, there are three so-called pretabular superintuitionistic logics (i.e. non-tabular, but all their own extensions are tabular). One of them – the logic \(L2\) – is characterized by the class of finite rooted linearly ordered sets of depth 2. It is established that for superintuitionistic logic \(L2\) in the language with one additional constant there are exactly five P. S. Novikov complete extensions; their semantic description is given. In this paper we propose an explicit axiomatics for each of the five existing P. S. Novikov complete extensions of the superintuitionistic logic \(L2\) in the language containing an additional constant.