  The notion of initial ideal for an ideal of a polynomial ring appears in the
theory of Gr\"obner basis. Similarly to the initial ideals, we can define the
initial algebra for a subalgebra of a polynomial ring, or more generally of a
Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra
Analogue to Gr\"obner Bases for Ideals) basis. The initial algebra of a
finitely generated subalgebra is not always finitely generated, and no general
criterion for finite generation is known.
  The aim of this paper is to present a new class of finitely generated
subalgebras having non-finitely generated initial algebras. The class contains
a subalgebra for which the set of initial algebras is continuum, as well as a
subalgebra with finitely many distinct initial algebras.
