  A detailed quantitative analysis of the system $(ppe)$ placed in magnetic
field ranging from $0 - 4.414 \times 10^{13} G$ is presented. The present study
is focused on the question of the existence of the molecular ion $H_2^{+}$ in a
magnetic field. As a tool, a variational method with an optimization of the
form of the vector potential (optimal gauge fixing) is used. It is shown that
in the domain of applicability of the non-relativistic approximation the system
$(ppe)$ in the Born-Oppenheimer approximation has a well-pronounced minimum in
the total energy at a finite interproton distance for $B \lesssim 10^{11} G$,
thus manifesting the existence of $H_2^{+}$. For $B \gtrsim 10^{11} G$ and
large inclinations (of the molecular axis with respect to the magnetic line)
the minimum disappears and hence the molecular ion $H_2^{+}$ does not exist. It
is shown that the most stable configuration of $H_2^{+}$ always corresponds to
protons situated along the magnetic line. With magnetic field growth the ion
$H_2^{+}$ becomes more and more tightly bound and compact, and the electronic
distribution evolves from a two-peak to a one-peak pattern. The domain of
inclinations where the $H_2^{+}$ ion exists reduces with magnetic field
increase and finally becomes $0^o - 25^o$ at $B = 4.414 \times 10^{13} G$.
Phase transition type behavior of variational parameters for some interproton
distances related to the beginning of the chemical reaction $H_2^{+}
\leftrightarrow H + p$ is found.
