Initial data for a head-on collision of two Kerr-like black holes with close limit

We prove the existence of a family of initial data for the Einstein vacuum equation which can be interpreted as the data for two Kerr-like black holes in an arbitrary location and with spins pointing in an arbitrary direction. This family of initial data has the following properties: (i) When the mass parameter of one of them is zero or when the distance between them goes to infinity, it reduces exactly to the Kerr initial data; (ii) when the distance between them is zero, we obtain exactly the Kerr initial data with mass and angular momentum equal to the sum of the mass and angular momentum parameters of each of them. The initial data depend smoothly on the distance, the mass, and the angular momentum parameters.