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We study finite systems of subspaces of a complex Hilbert space such thateach pair of subspaces satisfies a certain condition as described in thefollowing. For each subspace excepting the first one an angle between thissubspace and the first one is fixed. The set of all subspaces excluding thefirst one is divided into disjoint subsets. Each such subset consists of one ortwo subspaces. Subspaces from different subsets are orthogonal and orthogonalprojections onto subspaces from the same subset commute.

On systems of subspaces of a Hilbert space such that every pair of subspaces satisfies one of the angle or commutativity condition