  In this paper two things are done. First, it is pointed out the existence of
exact asymptotically flat, spherically symmetric black holes when a self
interacting, minimally coupled scalar field is the source of the energy
momentum of the Einstein equations in four dimensions. The scalar field
potential is the recently found to be compatible with the hairy generalization
of the Plebanski-Demianski solution of general relativity. This paper describes
the spherically symmetric solutions that smoothly connect the Schwarzschild
black hole with its hairy counterpart. The geometry and scalar field are
everywhere regular except at the usual Schwarzschild like singularity inside
the black hole. The scalar field energy momentum tensor satisfies the null
energy condition in the static region of the spacetime. The first law holds
when the parameters of the scalar field potential are fixed under
thermodynamical variation. Secondly, it is shown that an extra, dimensionless
parameter, present in the hairy solution, allows to modify the gravitational
field of a spherically symmetric black hole in a remarkable way. When the
dimensionless parameter is increased, the scalar field generates a flat
gravitational potential, that however asymptotically matches the Schwarzschild
gravitational field. Finally, it is shown that a positive cosmological constant
can render the scalar field potential convex if the parameters are within a
specific rank.
