  In this note, we make a step towards the classification of toric surfaces
admitting reducible Severi varieties. We generalize the results of [Lan19,
Tyo13, Tyo14], and provide two families of toric surfaces admitting reducible
Severi varieties. The first family is general, and is obtained by a quotient
construction. The second family is exceptional, and corresponds to certain
narrow polygons, which we call kites. We introduce two types of invariants that
distinguish between the components of the Severi varieties, and allow us to
provide lower bounds on the numbers of the components. The sharpness of the
bounds is verified in some cases, and is expected to hold in general for ample
enough linear systems. In the appendix, we establish a connection between the
Severi problem and the topological classification of univariate polynomials.
