In the present paper we consider a family of <$E2~times~2> operator matrices <$EA sub mu (k)>, <$Ek~symbol <174>~roman T sup 3 ~:=~(- pi ,~pi ] sup 3>, <$Emu~>>~0>, associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice Z<^>3, interacting via creation and annihilation operators. We prove that there is a value <$Emu sub 0> of the parameter <$Emu> such that only for <$Emu~=~mu sub 0> the operator <$EA sub mu ( 0 Bar )> has a virtual level at the point <$Ez~=~0~=~min~sigma sub ess (A sub mu ( 0 Bar ))> and the operator <$EA sub mu ( pi bar )> has a virtual level at the point <$Ez~=~18~=~max~sigma sub ess (A sub mu ( pi bar ))>, where <$E0 Bar~:=~(0,~0,~0)>, <$Epi bar~:=~( pi ,~pi ,~pi )~symbol <174>~roman T sup 3>. The absence of the eigenvalues of <$EA sub mu (k)> for all values of k under the assumption that <$Emu~=~mu sub 0> is shown. The threshold energy expansions for the Fredholm determinant associated to <$EA sub mu (k)> are obtained.

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