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two components of the total magnetic field: | B ext | | B $^{self}$| ∼ O ( ε k M ) , with k ≥ 1. This means that the self-field is the dominant component: the magnetic field is primar- ily self-generated. Also, the overall magnetic field is as- sumed to be slowly varying in time, i.e., to be of the form B ( r , ε$_{M}$ t ), while B self and B ext are defined as 

where B$_{T}$ ≡ I ( r , ε$_{M}$ t ) ∇ ϕ and B$_{P}$ ≡ ∇ ψ$_{p}$ ( r , ε$_{M}$ t ) × ∇ ϕ are the toroidal and poloidal components of the self-field, with ( ψ$_{p}$, ϕ, ϑ ) defining locally a set of magnetic coordi- nates. Moreover, the external magnetic field B ext is as- sumed to be purely poloidal and defined in terms of the vacuum potential ψ$_{D}$ ( r , ε$_{M}$ t ). In particular, we notice here that for typical astrophysical applications of inter- est, the function ψ$_{D}$ ( r , ε$_{M}$ t ) can be conveniently identi- fied with the flux function of a dipolar magnetic field. It follows that the magnetic field can also be written in the form 

where the function ψ ( r , ε$_{M}$ t ) is defined as ψ ( r , ε$_{M}$ t ) ≡ ψ$_{p}$ ( r , ε$_{M}$ t ) + ψ$_{D}$ ( r , ε$_{M}$ t ) , and ( ψ, ϕ, ϑ ) define a set of local magnetic coordinates, as implied by the equation B · ∇ ψ = 0 which is identically satisfied. In addi- tion, it is assumed that the charged particles of the plasma are subject to the action of effective EM poten- tials { Φ eff s ( r , ε$_{M}$ t ) , A ( r , ε$_{M}$ t ) } , where A ( r , ε$_{M}$ t ) is the vector potential corresponding to the magnetic field of Eq.(4), while Φ eff s ( r , ε$_{M}$ t ) is given by 

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with Φ eff s ( r , ε$_{M}$ t ) , Φ( r , ε$_{M}$ t ) and Φ$_{G}$( r , ε$_{M}$ t ) denoting the effective electrostatic potential and the electrostatic and generalized gravitational potentials (the latter, in principle, being produced both by the central object and the accretion disc). Finally, both the equilibrium effec- tive electric field E eff s , generated by the combined action of the effective EM potentials and defined as 

and the magnetic field B are also assumed to be axi- symmetric. 

## III. FIRST INTEGRALS OF MOTION AND GUIDING-CENTER ADIABATIC INVARIANTS 

In the present formulation, assuming axi-symmetry and stationary EM and gravitational fields, the exact first integrals of motion can be immediately recovered from the symmetry properties of the single charged par- ticle Lagrangian function L . In particular, these are the 

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total particle energy 

and the canonical momentum p$_{ϕs}$ (conjugate to the ig- norable toroidal angle ϕ ) 

Gyrokinetic theory allows one to derive the adiabatic in- variants of the system [28, 29]; by construction, these are quantities conserved only in an asymptotic sense, i.e., only to a prescribed order of accuracy. As is well known, gyrokinetic theory is a basic prerequisite for the investigation both of kinetic instabilities (see for example [32–34]) and of equilibrium flows occurring in magnetized plasmas [24, 35–38]. For astrophysical plasmas close to compact objects, this generally involves the treatment of strong gravitational fields which needs to be based on a covariant formulation (see [39–42]). However, for non-relativistic plasmas (in the sense already discussed), the appropriate formulation can also be directly recov- ered via a suitable reformulation of the standard (non- relativistic) theory for magnetically confined laboratory plasmas [29–31, 43–49]. In connection with this, con- sider again the Lagrangian function L of charged parti- cle dynamics. By performing a gyrokinetic transforma- tion of L , accurate to the prescribed order in ε$_{M}$ , it fol- lows that - by construction - the transformed Lagrangian L ′ becomes independent of the guiding-center gyrophase angle φ $^{′}$. Therefore, by construction, the canonical mo- mentum p ′ φ $^{′}$s = ∂ L $^{′}$/∂ · φ $^{′}$, as well as the related mag- netic moment defined as m ′ s ≡ Z$_{s}$e M$_{s}$$_{c}$p ′ φ ′ $_{s}$, are adiabatic invariants. As shown by Kruskal (1962 [50]) it is al- ways possible to determine L ′ so that m ′ s is an adi- abatic invariant of arbitrary order in ε$_{M}$ , in the sense that 1 Ω ′ cs d dt ln m ′ s = 0+ O ( ε n +1 M ) , where Ω ′ cs = Z$_{s}$eB $^{′}$/M$_{s}$ c denotes the Larmor frequency evaluated at the guiding- center and the integer n depends on the approximation used in the perturbation theory to evaluate m $^{′}$$_{s}$. In ad- dition, the guiding-center invariants corresponding to E$_{s}$ and ψ$_{∗}$$_{s}$ (denoted as E ′ s and ψ ′ ∗ s respectively) can also be given in terms of L $^{′}$. These are also, by definition, manifestly independent of φ $^{′}$., ′ 

This basic property of the magnetic moment m s is es- sential in the subsequent developments. Indeed, we shall prove that it allows the effects of temperature anisotropy to be included in the asymptotic stationary solution. 

Let us now define the concept of gyrokinetic and equi- librium KDFs. 

## Def. - Gyrokinetic KDF (GK KDF) 

A generic KDF f$_{s}$ ( r , v , t ) will be referred to as gy- rokinetic if its Lagrangian time-derivative d $_{dt}$f$_{s}$ ( r , v , t ) is independent of the gyrophase angle φ ′ evaluated the guiding-center position when its state x = ( r , v ) is espressed as a function of an arbitrary gyrokinetic state z $^{′}$= ( y $^{′}$, φ$^{′}$) . More generally, in the following f$_{s}$ ( r , v , t ) 

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