The Gateway cloning system utilizes what Life Technologies refers to as "Clonase enzyme mix" to catalyze the BP and LR reactions.
What is in this enzyme mix? Is there a sequence for them?
The prototypes for these proteins are:
the bacteriophage λ proteins Int and Xis, required for, respectively the integration and excision of prophages. They are, as mentioned by @WYSIWYG, recombinases.
Genetic map of bacteriophage λ (source of image)
the original Ihf (integrative/integration host factor) was identified as an E. coli protein required for prophage integration.
As you might expect Int and Xis proteins occur in many phage genomes, and in integrated prophages in bacterial genomes, and Ihf, because it is involved in bacterial DNA replication, is ubiquitous. I imagine the actual source of the proteins used by Life Technologies could be a trade secret.