Maximal partial spreads of polar spaces

Some constructions of maximal partial spreads of finite classical polar spaces are provided. In particular we show that, for n ≥ 1, H(4n−1, q2) has a maximal partial spread of size q2n+ 1, H(4n + 1, q2) has a maximal partial spread of size q2n+1+ 1 and, for n ≥ 2, Q+(4n-1, q), Q(4n-2, q), W(4n − 1, q), q even, W(4n-3, q), q even, have a maximal partial spread of size qn+ 1.