  A nonsingular rational curve $C$ in a complex manifold $X$ whose normal
bundle is isomorphic to $${\mathcal O}_{{\mathbb P}^1}(1)^{\oplus p} \oplus
{\mathcal O}_{{\mathbb P}^1}^{\oplus q}$$ for some nonnegative integers $p$ and
$q$ is called an unbendable rational curve on $X$. Associated with it is the
variety of minimal rational tangents (VMRT) at a point $x \in C,$ which is the
germ of submanifolds ${\mathcal C}^C_x \subset {\mathbb P} T_x X$ consisting of
tangent directions of small deformations of $C$ fixing $x$. Assuming that there
exists a distribution $D \subset TX$ such that all small deformations of $C$
are tangent to $D$, one asks what kind of submanifolds of projective space can
be realized as the VMRT ${\mathcal C}^C_x \subset {\mathbb P} D_x$. When $D
\subset TX$ is a contact distribution, a well-known necessary condition is that
${\mathcal C}_x^C$ should be Legendrian with respect to the induced contact
structure on ${\mathbb P} D_x$. We prove that this is also a sufficient
condition: we construct a complex manifold $X$ with a contact structure $D
\subset TX$ and an unbendable rational curve $C \subset X$ such that all small
deformations of $C$ are tangent to $D$ and the VMRT ${\mathcal C}^C_x \subset
{\mathbb P} D_x$ at some point $x\in C$ is projectively isomorphic to an
arbitrarily given Legendrian submanifold. Our construction uses the geometry of
contact lines on the Heisenberg group and a technical ingredient is the
symplectic geometry of distributions the study of which has originated from
geometric control theory.
