  In this paper we consider a family of generalized Korteweg-de Vries equations
and study the linear modulational instability of small amplitude traveling
waves solutions. Under explicit non-degeneracy conditions on the dispersion
relation, we completely describe the spectrum near the origin of the linearized
operator at such solutions and prove that the unstable spectrum (when present)
is composed by branches depicting always a closed figure ''8''. We apply our
abstract theorem to several equations such as the Whitham, the
gravity-capillary Whitham and the Kawahara equations, confirming that the
unstable spectrum of the corresponding linearized operators exhibits a figure
''8'' instability, as it was observed before only numerically. Our method of
proof uses a symplectic version of Kato's theory of similarity transformation
to reduce the problem to determine the eigenvalues of a $3 \times 3$ complex
Hamiltonian and reversible matrix. Then, via a block-diagonalization procedure,
we conjugate such matrix into a block-diagonal one composed by a $2\times 2$
Hamiltonian and reversible matrix, describing the unstable spectrum, and a
single purely imaginary element describing the stable eigenvalue.
