Suppose $X$ is projective manifold endowed with holomorphic symplectic form. Let $D$ be smooth divisor on $X$.

1. Characteristic foliation with algebraic and non-algebraic leaves. Suppose that $X$ is a product of two abelian $3$-folds $A_1 \times A_2$ in such a way that the fibers of the natural projection $\pi: X \to A_2$ are Lagrangian. Let $D$ be the pull-back under $\pi$ of an ample divisor $E$ on $A_2$. The characteristic foliation on $D$ will be everywhere tangent to $\pi$ and over each fiber is a linear foliation. The slope of the foliation on the fiber over a a point $p \in E$ is determined by the tangent of $E$ at $p$. But since $E$ is an ample divisor, it has non-degenerate Gauss map and every possible linear foliation on $A_1$ will appear among the restriction of the characteristic foliation $D$ to fibers of $\pi$. If $A_1$ itself is a product of elliptic curves then $A_1$ carries linear foliations with all leaves algebraic. This shows the existence of $5$-dimensional divisors with the requested property. Starting with a $4$-dimensional symplectic manifolds carrying a Lagrangian fibration one can adapt the above argument to produce divisors with similar behaviour if the Lagrangian fibration admits a fiber which is isogeneous to the product of elliptic curves.

2. Characteristic foliation with all leaves algebraic. If every leaf of the characteristic foliation is algebraic then it was proved by Amerik and Campana (refining previous result by Hwang and Viehweg) that either every leaf is rational or $X$ is, up to étale coverings, the product of a symplectic surface $S$ with a symplectic manifold $Y$ and $D$ is the product of a curve $C\subset S$ with $Y$. In the latter case the leaves of the characteristic foliation are the fibers of the projection to $Y$. In particular, the Kodaira dimension of $D$ is at most $1$.

3. Characteristic foliation on ample divisors. One interesting problem on the subject (already raised by Hwang and Viehweg) is whether or not the characteristic foliation on an ample divisor can have one algebraic leaf. To the best of my knowledge this problem is wide open.

Suppose $X$ is projective manifold endowed with holomorphic symplectic form. Let $D$ be smooth divisor on $X$.

1. Characteristic foliation with algebraic and non-algebraic leaves. Suppose that $X$ is a product of two abelian varieties of dimension $n$ ($n\ge 2$) $A_1 \times A_2$ in such a way that the fibers of the natural projection $\pi: X \to A_2$ are Lagrangian. Let $D$ be the pull-back under $\pi$ of an ample divisor $E$ on $A_2$. The characteristic foliation on $D$ will be everywhere tangent to $\pi$ and over each fiber is a linear foliation. The slope of the foliation on the fiber over a a point $p \in E$ is determined by the tangent of $E$ at $p$. But since $E$ is an ample divisor, it has non-degenerate Gauss map and every possible linear foliation on $A_1$ will appear among the restriction of the characteristic foliation $D$ to fibers of $\pi$. If $A_1$ itself is a product of elliptic curves (or isogeneous to the product of an elliptic curve and an abelian variety of dimension $n-1$) then $A_1$ carries linear foliations with all leaves algebraic. This shows the existence of divisors with the requested property.

2. Characteristic foliation with all leaves algebraic. If every leaf of the characteristic foliation is algebraic then it was proved by Amerik and Campana (refining previous result by Hwang and Viehweg) that either every leaf is rational or $X$ is, up to étale coverings, the product of a symplectic surface $S$ with a symplectic manifold $Y$ and $D$ is the product of a curve $C\subset S$ with $Y$. In the latter case the leaves of the characteristic foliation are the fibers of the projection to $Y$. In particular, the Kodaira dimension of $D$ is at most $1$.

3. Characteristic foliation on ample divisors. One interesting problem on the subject (already raised by Hwang and Viehweg) is whether or not the characteristic foliation on an ample divisor can have one algebraic leaf. To the best of my knowledge this problem is wide open.