  By proving the existence of non-monotone and non-oscillating wavefronts for
the Nicholson's blowflies diffusive equation (the NDE), we answer an open
question raised in [16]. Surprisingly, these wavefronts can be observed only
for sufficiently small delays. Similarly to the pushed fronts, obtained waves
are not linearly determined. In contrast, a broader family of eventually
monotone wavefronts for the NDE is indeed determined by properties of the
spectra of the linearized equations. Our proofs use essentially several
specific characteristics of the blowflies birth function (its unimodal form and
the negativity of its Schwarz derivative, among others). One of the key
auxiliary results of the paper shows that the Mallet-Paret--Cao--Arino theory
of super-exponential solutions for scalar equations can be extended for some
classes of second order delay differential equations. For the new type of
non-monotone waves to the NDE, our numerical simulations also confirm their
stability properties established by Mei et al.
