Pope_Leo_XIIIHoly Mary, Mother of Unity
Pope Leo XIII requested that the Rosary be prayed specifically for the intention of reconciliation and reunion. Here is the relevant section from his Encyclical of September 20 1896, Fidentem Piumque Animum:

The Power of Prayer
That earnest desire, which We have learnt from the Divine Heart of Jesus, of fostering the work of reconciliation among those who are separated from Us daily urges Us more pressingly to action; and we are convinced that this most excellent Re-union cannot be better prepared and strengthened than by the power of prayer. The example of Christ is before us, for in order that His disciples might be one in faith and charity, he poured forth prayer and supplication to His Father.

The Patroness and Most Excellent Custodian of Unity
And concerning the efficacious prayer of His most holy Mother for the same end, there is a striking testimony in the Acts of the Apostles. Therein is described the first assembly of the Disciples, expecting with earnest hope and prayer the promised fullness of the Holy Spirit. And the presence of Mary united with them in prayer is specially indicated: All these were persevering with one mind in prayer with Mary the Mother of Jesus (Acts 1:14). Wherefore as the nascent church rightly joined itself in prayer with her as the patroness and most excellent custodian of Unity, so in these times is it most opportune to do the same all over the Catholic World.

Nothing More Acceptable to Mary
Let then the zeal for this prayer (of the Rosary) everywhere be re-kindled, particularly for the end of Holy Unity. Nothing will be more agreeable and acceptable to Mary; for, as she is most closely united with Christ she especially wishes and desires that they who have received the same Baptism with Him may be united with Him and with one another in the same faith and perfect charity. So may the sublime mysteries of this same faith by means of the Rosary devotion be more deeply impressed in men’s minds, with the happy result that “we may imitate what they contain and obtain what they promise”.