  Starting from essentially commutative exponential map $E(B|I)$ for generic
tensor-valued 2-forms $B$, introduced in \cite{Akh} as direct generalization of
the ordinary non-commutative $P$-exponent for 1-forms with values in matrices
(i.e. in tensors of rank 2), we suggest a non-trivial but multi-parametric
exponential ${\cal E}(B|I|t_\gamma)$, which can serve as an interesting
multi-directional evolution operator in the case of higher ranks. To emphasize
the most important aspects of the story, construction is restricted to
backgrounds $I_{ijk}$, associated with the structure constants of {\it
commutative} associative algebras, what makes it unsensitive to topology of the
2d surface. Boundary effects are also eliminated (straightfoward generalization
is needed to incorporate them).
