  We show that any smooth surface germ in the moduli of abelian surfaces arises
from a Lagrangian fibration of abelian surfaces. By Donagi-Markman's cubic
condition, the key issue of the proof is to find a suitable affine structure
with a compatible cubic form on the base space of the family. We achieve this
by analyzing the properties of cubic forms in two variables and proving the
existence of the solution of the resulting partial differential equations by
Cauchy-Kowalewski Theorem. Modifying the argument, we show also that a smooth
curve germ in the moduli of abelian surfaces arises from a Lagrangian fibration
if and only if the curve is a null curve with respect to the natural
holomorphic conformal structure on the moduli of abelian surfaces.
