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We have found four different types of one-half monopole charge solutions that we label as Type $A1$, $A2$, $B1$, and $B2$. The Type $B$ ($B1$ and $B2$) solutions seem to be a $180^o$ rotation of the $z$-axis about the origin, $r=0$, of the Type $A$ ($A1$ and $A2$) solutions. However for a particular value of Higgs self-coupling constant, $\lambda$, the Type 1 ($A1$ and $B1$) and Type 2 ($A2$ and $B2$) solutions possess different total energies as well as energy densities distributions. They also possess different finite Lagrangian and different Lagrangian density distribution in three space.

The gauge potentials of the Type $A$ solutions are singular at only one point at infinity on the negative $z$-axis whereas the Type $B$ solutions are singular at only one point at infinity on the positive $z$-axis. The 't Hooft magnetic fields of these solutions at spatial infinity correspond to the magnetic field of a positive one-half monopole located at the origin, $r=0$, and a semi-infinite Dirac string singularity located on one-half of the $z$-axis which carries a magnetic flux of $\frac{2\pi}{g}$ going into the center of the sphere at infinity. Hence the net magnetic charge of the configuration is zero. These solutions do not satisfy the first order Bogomol'nyi equations and are non-BPS solutions even in the BPS limit of vanishing Higgs self-coupling as their Lagrangian densities do not vanish over all space and their Lagrangians are finite quantities when $\lambda=0$. When $\lambda=0$, the dimensionless total energy of the one-half monopole located at $r=0$ of the Type 1 solutions is 0.51 and the that of the Type 2 solutions is 0.53. Hence the total energies of these one-half monopole solutions are larger than the dimensionless total energy of the BPS solution which is 0.50. The total energies of these one-half monopole solutions are calculated for various strength of the Higgs field self-coupling constant $\lambda$ from zero to 100 and they are found to increase logarithmically with $\lambda$.