  A thorough study is performed of the analytical properties of the fermion
determinant for the case that the time components of (axial) vector fields do
not vanish. For this purpose the non--Hermitian Euclidean Dirac Hamiltonian is
generalized to the whole complex plane. The Laurent series are proven to reduce
to Taylor series for the corresponding eigenvalues and --functions as long as
field configurations are assumed for which level crossings do not occur. The
condition that no level crossings appears determines the radius convergence.
However, the need for regularization prohibits the derivation of an analytic
energy functional because real and imaginary parts of the eigenvalues are
treated differently. Consistency conditions for a Minkowski energy functional
are extracted from global gauge invariance and the current field identity for
the baryon current. Various treatments of the Nambu--Jona--Lasinio soliton are
examined with respect to these conditions. Motivated by the studies of the
Laurent series for the energy functional the Euclidean action is expanded in
terms of the $\omega$--field. It is argued that for this expansion the
proper--time regularization scheme has to be imposed on the operator level
rather than on an expression in terms of the one--particle eigenenergies. The
latter treatment is plagued by the inexact assumption that the Euclidean Dirac
Hamiltonian and its Hermitian conjugate can be diagonalized simultaneously. It
is then evident that approaches relying on counting powers of the
$\omega$--field in the one--particle eigenenergies are
