Unification Problem in Nelson's Logic $\mathbf{N4}$

We consider the unification problem for formulas with coeffi-cients in the Nelson's paraconsitent logic $\mathbf{N4}$. By presence coefficients (parameters) the problem is quite not trivial and challenging (yet what makes the problem for $\mathbf{N4}$ to be peculiar is missing of replacement equivalents rule in this logic). It is shown that the unification problem in $\mathbf{N4}$ is decidable for $\sim$-free formulas. We also show that there is an algorithm which computes finite complete sets of unifiers (so to say — all best unifiers) for unifiable in $\mathbf{N4}$ $\sim$-free formulas (i.e. any unifier is equivalent to a substitutional example of a unifier from this complete set). Though the unification problem for all formulas (not $\sim$-free formulas) remains open.