The seniority symmetry is known to be partially conserved in two special cases in the (9/2)(4) system, which can lead to striking features in the corresponding structure and electromagnetic transition properties. However, it is still quite difficult to derive those kind of solvable states, in general, especially towards higher- j orbits. We have developed a novel and effective way to confront this challenge by starting from the m scheme and making use of the angular momentum projection method. It also allows us to explore another special family of seniority conserving states in the midshell besides the specific case of the (9/2)(4) configuration. Moreover, we have studied systemically all states in single- j systems up to j = 15/2 and derived the analytic expressions for the eigenvalues of all solvable states with a focus on those in j = 9/2 and j = 11/2. Such studies can also be useful for the experimental search of relevant states and for the understanding of their electromagnetic transition properties.