  Let $(M,I, \Omega)$ be a holomorphically symplectic manifold equipped with a
holomorphic Lagrangian fibration $\pi:\; M \mapsto X$, and $\eta$ a closed form
of Hodge type (1,1)+(2,0) on $X$. We prove that $\Omega':=\Omega+\pi^* \eta$ is
again a holomorphically symplectic form, for another complex structure $I'$,
which is uniquely determined by $\Omega'$. The corresponding deformation of
complex structures is called "degenerate twistorial deformation". The map $\pi$
is holomorphic with respect to this new complex structure, and $X$ and the
fibers of $\pi$ retain the same complex structure as before. Let $s$ be a
smooth section of of $\pi$. We prove that there exists a degenerate twistorial
deformation $(M,I', \Omega')$ such that $s$ is a holomorphic section.
