  Within the framework of the Duffin-Kemmer-Petiau (DKP) formalism a more
consistent approach to the derivation of the third order wave equation obtained
earlier by M. Nowakowski [Phys.Lett.A {\bf 244} (1998) 329] on the basis of
heuristic considerations is suggested. For this purpose an additional algebraic
object, the so-called $q$ - commutator ($q$ is a primitive cubic root of unity)
and a new set of matrices $\eta_{\mu}$ instead of the original matrices
$\beta_{\mu}$ of the DKP algebra are introduced. It is shown that in terms of
these $\eta_{\mu}$ matrices we have succeeded in reducing a procedure of the
construction of cubic root of the third order wave operator to a few simple
algebraic transformations and to a certain operation of the passage to the
limit $z \rightarrow q$, where $z$ is some complex deformation parameter
entering into the definition of the $\eta$ - matrices. A corresponding
generalization of the result obtained to the case of the interaction with an
external electromagnetic field introduced through the minimal coupling scheme
is carried out and a comparison with M. Nowakowski's result is performed. A
detailed analysis of the general structure for a solution of the first order
differential equation for the wave function $\psi(x; z)$ is performed and it is
shown that the solution is singular in the $z \rightarrow q$ limit. The
application to the problem of construction within the DKP approach of the path
integral representation in parasuperspace for the propagator of a massive
vector particle in a background gauge field is discussed.
