  We study exact, analytic, static, spherically symmetric, four-dimensional
solutions of minimally coupled Einstein-scalar gravity, sourced by a scalar
field whose profile has the form of the sine-Gordon soliton. We present a
horizonless, everywhere regular and positive-mass solution (a solitonic star)
and a black hole. The scalar potential behaves as a constant near the origin
and vanishes at infinity. In particular, the solitonic scalar star interpolates
between an anti-de Sitter and an asympototically flat spacetime. The black-hole
spacetime is unstable against linear perturbations, while due to numerical
issues, we were not able to determine with confidence whether or not the
star-like background solution is stable.
