Two types of boson stars in the ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> )-dimensional <mml:math xmlns:mml="http://www.w3.

We investigate boson stars in an <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>O</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>3</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> scalar field theory with a symmetry-breaking potential. By constructing numerically spherically symmetric solutions, we demonstrate that the model gives rise to a rich set of field configurations. The negative coupling constant of the scalar self-interactions allows for two types of boson stars: type I solutions represent the usual boson stars, which emerge from the vacuum as the boson frequency is decreased below the boson mass, whereas type II boson stars emerge from a set of static soliton solutions. Depending on the strength of the gravitational coupling constant, both types, or only one type, are present. At a critical set of coupling constants, both types undergo a bifurcation. There the spirals of both types disconnect from their branches and reconnect with each other, while the remaining branches of both types also reconnect with each other. Analogous features are seen for U(1)-gauged boson stars.