**abstract:**
The three-state Grover walk on a line exhibits the localization effect. It is characterized by a non-vanishing probability of the particle to stay at the origin. We present two continuous deformations of the Grover walk which preserve its localization nature. The resulting quantum walks differ in the rate at which they spread through the lattice. The analogy with the wave propagation allows us to adopt many of its concepts. For instance the dispersion relation and the group velocity. The velocities of the left and right-travelling probability peaks are then given by the maximum of the group velocity. We find the explicit form of peak velocities in dependence on the coin parameter. Our results show that the localization of the quantum walk is not a singular property of an isolated coin but can be found for entire families of coins. The presented constructions can be extended to higher-dimensional quantum walk in a straightforward way.

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