Multiple kink solutions for M-component Burgers equations in (1+1)-dimensions and (2+1)-dimensions

M-component Burgers equations in (1+1)-dimensions and (2+1)-dimensions are examined for complete integrability. The Cole-Hopf transformation method and the simplified form of Hereman's method are used to achieve this goal. Multiple kink solutions and multiple singular kink solutions are formally derived for each vector equation.