I believe the conditions are that the poisson bracket of $C_1$ and $C_2$ is $\ne 0$ and the differentials $dC_1$ and $dC_2$ are linearly independent at all points of $\mathcal{N}$. The second condition implies that N is a submanifold. If the first condition is not fulfilled, $N$ is clearly not a sympletic submanifold because we could take the first two coordinates in the Darboux theorem to be $C_1,C_2$. If the first condition is fulfilled, we take the coordinate system such that the first two coordinates are $C_1, C_2$. In this coordinates the matrix of the symplectic form is blockdiagonal with the first $2\times 2$ block being nondegenerate matrix and all other elements such that the index contains 2'' equal to $0$. Then, the second block is nondegenerate and it is more or less the same as the matrix of the restriction of the form to $\mathcal{N}$
I believe the conditions are that the poisson bracket of $C_1$ and $C_2$ is $\ne 0$ and the differentials $dC_1$ and $dC_2$ are linearly independent at all points of $\mathcal{N}$. The second condition implies that N is a submanifold. If the first condition is not fulfilled, $N$ is clearly not a sympletic submanifold because we could take the first two coordinates in the Darboux theorem to be $C_1,C_2$. If the first condition is fulfilled, we take the coordinate system such that the first two coordinates are $C_1, C_2$. In this coordinates the matrix of the symplectic form is blockdiagonal with the first $2\times 2$ block being nondegenerate matrix and all elements such that the index contains 2'' equal to $0$. Then, the second block is nondegenerate and it is more or less the same as the matrix of the restriction of the form to $\mathcal{N}$