  Gel'fand-Dorfman bialgebra, which is both a Lie algebra and a Novikov algebra
with some compatibility condition, appears in the study of Hamiltonian pairs in
completely integrable systems and a class of special Lie conformal algebras
called quadratic Lie conformal algebras. In this paper, we investigate the
extending structures problem for Gel'fand-Dorfman bialgebras, which is
equivalent to some extending structures problem of quadratic Lie conformal
algebras. Explicitly, given a Gel'fand-Dorfman bialgebra $(A, \circ,
[\cdot,\cdot])$, this problem asks that how to describe and classify all
Gel'fand-Dorfman bialgebraic structures on a vector space $E$ $(A\subset E$)
such that $(A, \circ, [\cdot,\cdot])$ is a subalgebra of $E$ up to an
isomorphism whose restriction on $A$ is the identity map. Motivated by the
theories of extending structures for Lie algebras and Novikov algebras, we
construct an object $\mathcal{GH}^2(V,A)$ to answer the extending structures
problem by introducing a definition of unified product for Gel'fand-Dorfman
bialgebras, where $V$ is a complement of $A$ in $E$. In particular, we
investigate the special case when $\text{dim}(V)=1$ in detail.
