You can't call it a separation. Rama abandons Lakshmana. He says abandoning good people is as good as killing them.
विसर्जये त्वां सौमित्रे मा भूद् धर्मविपर्ययः ।
त्यागो वधो वा विहितः साधूनां ह्युभयं समम् ॥ १३ ॥
Story: While Rama is having a secret meeting with Devatas, for it to be uninterrupted, he gives this responsibility to Lakshmana. If he fails he was to be killed. But due to Durvasa's entry Lakshmana could not keep his responsibility. So Rama in great grieve says that he has abandoned him. Lakshmana goes to Sarayu river doing achamana, he holds his breath and ends his avatara.