We study differential equations that lead to extremal points in symplectic pseudospectra. In a two-level approach, where on the inner level we compute extremizers of the symplectic epsilon-pseudospectrum for a given epsilon and on the outer level we optimize over epsilon, this is used to solve symplectic matrix nearness problems such as the following: For a symplectic matrix with eigenvalues of unit modulus, we aim to determine the nearest complex symplectic matrix such that some or all eigenvalues leave the complex unit circle. Conversely, for a symplectic matrix with all eigenvalues lying off the unit circle, we consider the problem of computing the nearest symplectic matrix that has an eigenvalue on the unit circle.