  First, the creation of a free neutron spinless pairs is predicted in a
superfluid liquid $^4$He and a neutron gas mixture. For solving the given
problem, it is presented an exact solution to the model of dilute Bose gas as
an extension of the Bogoliubov model, at quantity of the condensate fraction
varying in the state $0\leq\frac{N_0}{N}\leq 1$, which in turn might be useful
for a description of the superfluid liquid $^4$He. Due to an application of
presented new model of dilute Bose gas, we prove that an appearance of atoms in
the condensate is a suppressor for the collective modes as well as a creator
for single-particle excitations On other hand, it is shown that the terms of
the interaction between the Bogoliubov excitations (Bogoliubov phonon-roton
modes) and the density modes of the neutron meditate an attractive interaction
via the neutron modes, which in turn leads to a bound state on a spinless
neutron pair.
