Per the Whitney embedding theorem, any manifold $M$ can be embedded into a sufficiently high dimensional Euclidean space.

According to Gromov's h-principle for contact embeddings, any contact manifold admits high codimension embeddings contact embeddings into the standard contact $\mathbb{R}^{2n+1}$, so in particular any exact symplectic manifold can be thusly embedded. It is also known that integral symplectic manifold can be embedded into a sufficiently high dimensional projective space.

Meanwhile in complex geometry, certainly not all complex manifolds embed into $\mathbb{C}^N$ for any $N$, or for that matter (as far as I know) into any particular space. But large classes do; the Stein manifolds into $\mathbb{C}^N$, and the integral Kahler manifolds into projective space.

Which classes of holomorphic symplectic manifolds admit (high codimension) embeddings as holomorphic symplectic submanifolds of some standard holomorphic symplectic space?

I would particularly like to know:

When can a holomorphic symplectic manifold $W$ be embedded as a (high codimension) holomorphic symplectic submanifold of the holomorphic contact manifold $\mathbb{P} T^* M$ for some complex manifold $M$?

By a symplectic submanifold of a contact manifold, I mean that for some choice of the contact form $\lambda$, the restriction of $d \lambda$ gives the symplectic form.

For this question I allow $M$ to be arbitrary (e.g. you may take $M = W$), so the complex geometry itself should give no obstructions.

I am particularly interested in complex symplectic manifolds which are conic in the sense of admitting a $\mathbb{C}^*$ action which scales the symplectic form. These are complex analogues of the exact symplectic manifolds mentioned above, since a $\mathbb{R}$ action scaling the symplectic form is a vector field $Z$ with $\omega = Z \omega = d i_Z \omega + i_Z d \omega = d \omega (Z, \cdot)$.