In this talk, we are concerned with the initial-value problem for the modified two-component Euler-Poincar\'{e} equations including the classical Euler-Poincar\'{e} equations, integrable two-component Camassa-Holm system and its two-component modified version. We first establish the optimal local well-posedness and blow-up criteria for strong solutions to the equations in the Besov spaces. Then we construct its global and blow-up strong solutions by using the orthogonal and symmetric transform invariances. Subsequently, we show rigorously that the equations will recover to a symmetric hyperbolic system of conservation laws as the dispersion parameters vanish. Finally, we prove the Liouville-type theorem for the stationary weak solutions to the equations. This is a joint work with Professor Yue Liu from the University of Texas at Arlington, USA.