The electrostatics of two cylinders charged to the symmetrical or anti-symmetrical potential is studied by using the null-field boundary integral equation (BIE) in conjunction with the degenerate kernel in terms of bipolar coordinates. The undetermined coefficient is obtained according to the Fredholm alternative theorem. Uniqueness of solution, infinite solution and no solution is therein discussed. A single cylinder case (circle or ellipse) is also provided for comparison. The link to the general solution space is also done. The condition at infinity is also correspondingly examined. The flux equilibrium along the circular boundaries and the infinite boundary is also checked as well as the contribution of the boundary integral (single and double layer potential) along the infinite boundary in the BIE is addressed. The ordinary and degenerate scales in the BIE are both investigated. Solution space limited by the BIE is also explained after comparing with the general solution. Identities between the present result and those in the literature of Darevski and Lekner results in Russia and NewZealand, respectively, is also constructed.