Checkpoints in searching for rational solutions of linear ordinary difference and differential systems

It is quite common that search algorithms for those solutions of difference and differential equations and systems that belong to a fixed class of functions are designed so that nonexistence of solutions of the desired type is detected only in the last stages of the algorithm. However, performing additional tests on the intermediate results makes it possible to stop the algorithm as soon as these tests imply that no solutions of the desired type exist. This gives an opportunity to save time and other computing resources. So, it makes sense to equip algorithms with checkpoints and some tests. We consider these questions in connection with the search for rational solutions of linear homogeneous difference and differential systems with polynomial coefficients, and propose a scheme equipped with such checkpoints and tests, and also report results of experiments with our implementation of the scheme in Maple.