The exact solutions to two-dimensional Yang-Mills theory: $su(2)$ and $u_\star(1)$

The purpose of this study is to show that the exact solutions to Yang-Mills theory can occur in two-dimensional space-time. We show that the instability of the stationary solutions for the nonabelian gauge field theories questioned by the dimensions of space-time. We show that instability of the stationary solution for the nonabelian gauge field theory questioned by the dimension of space-time. Another intention of this work is research of the source-free solutions to Maxwell's equations in a two-dimensional noncommutative temporal state space-time. We will show that there is comprehensively similarity between the existence of the bound states of nonabelian gauge theory and $u_\star(1)$.