The problem of determining time-independent solutions of the classical Yang-Mills equations for infinitely long charge cylinders is studied. A useful expression for the total energy in the field in terms of just the sources is derived. Numerical solutions have been found in the special cases of a small charge cylinder with a magnetic field B that either lies along the axis of symmetry or encircles the axis. It is as if these two solutions were due to currents encircling the axis or paralleling it, respectively. The condition that the solution behave well at infinity implies an exponential fall off for the fields in the azimuthal B field case and a fall off more rapid than 1/ R in the axial B field case, so that in both cases the existence of a B field requires the charge on the axis to be shielded. Consequently, these solutions do not behave at infinity at all like the Maxwell solution for a charge cylinder, and they have a lower energy per unit length. They show that in Yang-Mills theories the source does not determine a unique field. A classical interpretation of this is that the field remembers how the charges were transported during the construction of the cylinder. It also suggests that a quantum mechanical version of this problem would exhibit a “spontaneous symmetry breaking” to a less symmetric, lower energy vacuum. These solutions exhibit a twofold degeneracy, as the magnetic field may be either left- or right-handed in the azimuthal B field case, or point along the + z or − z axis in the axial B field case.