For the anisotropic $[u (\sum_{i=1}^N {\phi}_i^2)^2+v \sum_{i=1}^N \phi_i^4]$ field theory with \mbox{$N=2,3$} we calculate theimaginary parts of the renormalization-group functionsin the form of a series expansion in $v$, \mbox{i.\ e.,}around the isotropic case.Dimensional regularization is used to evaluate the fluctuationdeterminants for the isotropic instanton near the space dimension $4$.The vertex functions in the presence of instantons are renormalized with thehelp of a nonperturbative procedure introduced forthe simple $g{\phi}^4$-theory by McKane et al.