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To any Anosov flow X on a 3-manifold Fe1 associated a bi-foliated plane (aplane endowed with two transverse foliations Fs and Fu) which reflects thenormal structure of the flow endowed with the center-stable and center unstablefoliations. A flow is R-covered if Fs (or Fu) is trivial.  From one Anosov flow one can build infinitely many others byDehn-Goodman-Fried surgeries. This paper investigates how these surgeriesmodify the bi-foliated plane.  We first noticed that surgeries along some specific periodic orbits do notmodify the bi-foliated plane: for instance,  - surgeries on families of orbits corresponding to disjoint simple closedgeodesics do not affect the bi-foliated plane associated to the geodesic flowof a hyperbolic surface (Theorem 1);  - Fe2 associates a (non-empty) finite family of periodic orbits, calledpivots, to any non-R-covered Anosov flow. Surgeries on pivots do not affect thebranching structure of the bi-foliated plane (Theorem 2)  We consider the set Surg(A) of Anosov flows obtained by Dehn-Goodman-Friedsurgery from the suspension flows of Anosov automorphisms A in SL(2,Z) of thetorus T2.  Every such surgery is associated to a finite set of couples (C,m(C)), wherethe C are periodic orbits and the m(C) integers. When all the m(C) have thesame sign, Fenley proved that the induced Anosov flow is R-covered and twistedaccording to the sign of the surgery. We analyse here the case where thesurgeries are positive on a finite set X and negative on another set Y.  Among other results, we show that given any flow X in Surg(A) :  - there exists e>0 such that for every e-dense periodic orbit C, every flowobtained from X by a non trivial surgery along C is R-covered (Theorem 4).  - there exist periodic orbits C+,C- such that every flow obtained from X bysurgeries with distinct signs on C+ and C- is non-R-covered (Theorem 5).

Anosov flows on $3$-manifolds: the surgeries and the foliations