  In this paper we give the complete classification of solitons for a cubic NLS
equation on the simplest network with a non-trivial topology: the tadpole
graph, i.e. a ring with a half-line attached to it and free boundary conditions
at the junction. The model, although simple, exhibits a surprisingly rich
behavior and in particular we show that it admits: 1) a denumerable family of
continuous branches of embedded solitons bifurcating from linear eigenstates
and threshold resonances of the system; 2) a continuous branch of edge solitons
displaying a pitchfork symmetry breaking bifurcation at the threshold of the
continuous spectrum; 3) a finite family of continuous branches of solitons
without linear analogue. All the solutions are explicitly constructed in terms
of elliptic Jacobian functions. Moreover we show that families of nonlinear
bound states of the above kind continue to exist in the presence of a uniform
magnetic field orthogonal to the plane of the ring when a well definite flux
quantization condition holds true. Finally we highlight the role of resonances
in the linearization as a signature of the occurrence of bifurcations of
solitons from the continuous spectrum.
