In this talk, we shall propose the Gaussian mixture optimal transport Ensemble Kalman filter (GM-OT-EnKF) first. Then, we shall make two key observations, which can adapt the original GM-OT-EnKF more efficient to the partially linear systems. The one observation is the equivalence between the OT-EnKF with the classical Kalman filter (KF) in the linear system with Gaussian initial distribution. The other one is that in the setting of linear system with GM initial distribution, the updating of the components' weights in GM-OT-EnKF is finer than those in the Gaussian sum filter (GSF), due to the flexibility induced by the particles. These two observations in the linear system suggest two adaptions to the original GM-OT-EnKF corresponding to partially linear systems. The one is when the states' equation is linear, the EM algorithm is unnecessary in every cycles; the other one is when the observation is linear, the posterior mean and covariance matrix should be updated explicitly, rather than the empirical ones. The GM-OT-EnKF with either one of the two adaptions above is called the compact GM-OT-EnKF in this talk. The efficiency and accuracy of the GM-OT-EnKF and its compact version have been numerically verified in the estimation of the states in the Lorenz 63 system and the prediction of the remaining useful life of the lithium-ion batteries.