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We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works by D. Guan and the 1st author. Any such manifold   $Q$   of dimension   $2n-2$   is obtained as a finite degree   $n^2$   cover of some non-Kähler manifold   $W_F$  , which we call the  base  of   $Q$  . We show that the algebraic reduction of   $Q$   and its base is the projective space of dimension   $n-1$  . Besides, we give a partial classification of submanifolds in   $Q$  , describe the degeneracy locus of its algebraic reduction and prove that the automorphism group of   $Q$   satisfies the Jordan property.

Geometry and Automorphisms of Non-Kähler Holomorphic Symplectic Manifolds