Block-separation of variables: A form of partial separation for natural hamiltonians

We study twisted products H = αrHr of natural autonomous Hamiltonians Hr, each one depending on a separate set, called here separate r-block, of variables. We show that, when the twist functions αr are a row of the inverse of a block-Stäckel matrix, the dynamics of H reduces to the dynamics of the Hr, modified by a scalar potential depending only on variables of the corresponding r-block. It is a kind of partial separation of variables. We characterize this block-separation in an invariant way by writing in block-form classical results of Stäckel separation of variables. We classify the block-separable coordinates of 3.