  We demonstrate the systematic derivation of a class of discretizations of
nonlinear Schr{\"o}dinger (NLS) equations for general polynomial nonlinearity
whose stationary solutions can be found from a reduced two-point algebraic
condition. We then focus on the cubic problem and illustrate how our class of
models compares with the well-known discretizations such as the standard
discrete NLS equation, or the integrable variant thereof. We also discuss the
conservation laws of the derived generalizations of the cubic case, such as the
lattice momentum or mass and the connection with their corresponding continuum
siblings.
