State representation learning (SRL) in partially observable Markov decision processes has been studied to learn abstract features of data useful for robot control tasks. For SRL, acquiring domain-agnostic states is essential for achieving efficient imitation learning. Without these states, imitation learning is hampered by domain-dependent information useless for control. However, existing methods fail to remove such disturbances from the states when the data from experts and agents show large domain shifts. To overcome this issue, we propose a domain-adversarial and conditional state space model (DAC-SSM) that enables control systems to obtain domain-agnostic and task- and dynamics-aware states. DAC-SSM jointly optimizes the state inference, observation reconstruction, forward dynamics, and reward models. To remove domain-dependent information from the states, the model is trained with domain discriminators in an adversarial manner, and the reconstruction is conditioned on domain labels. We experimentally evaluated the model predictive control performance via imitation learning for continuous control of sparse reward tasks in simulators and compared it with the performance of the existing SRL method. The agents from DAC-SSM achieved performance comparable to experts and more than twice the baselines. We conclude domain-agnostic states are essential for imitation learning that has large domain shifts and can be obtained using DAC-SSM.