  We generalize Breuil-Hellmann-Schraen's local model for the trianguline
variety to certain points with non-regular Hodge-Tate weights. With the local
models we are able to prove, under the Taylor-Wiles hypothesis, the existence
of certain companion points on the global eigenvariety and the appearance of
related companion constituents in the completed cohomology for non-regular
crystalline Galois representations. The new ingredients in the proof of the
global applications are results relating the partial classicality of locally
analytic representations (the existence of non-zero locally algebraic vectors
in the non-Borel parabolic Emerton's Jacquet modules), the partially de Rham
properties of Galois representations (the de Rhamness of graded pieces along
the paraboline filtrations of the associated $(\varphi,\Gamma)$-modules over
the Robba rings) and the relevant properties of cycles on the generalized
Steinberg varieties. We prove that partial classicality implies partial de
Rhamness in finite slope cases using Ding's partial eigenvarieties.
