How to handle fixed values for variables in pre-processing
By definition, if these columns or features contain a constant value and yet the output variables change, then they are not influencing the output and likely can be ignored.
A more formal test is to determine how much of the variance between a model that uses that feature is attributable to that feature.
A simple example to illustrate this principle is to look up examples of PCA. In those examples, the technique tries and identifies feature that drive the most variance.