Under the strengthened Legendre condition, for any $x_0 \in [a,b]$ and any $c_0, c_1 \in \mathbb{R}$, there exists a unique Jacobi field $u \in C^2([a,b])$ satisfying $u(x_0) = c_0$ and $u'(x_0) = c_1$. In particular, the Jacobi fields form a two-dimensional vector space.