How do you translate from separate vector potential gradients to magnetic field components?

In an attempt at calculating anisotropic London penetration depths, I have obtained equations of the form

$$
begin{split}
-mu_0^{-1}nabla^2B_x &= Zpartial_yA_z – Tpartial_zA_x-Ypartial_zA_y,
-mu_0^{-1}nabla^2B_y &= Xpartial_xA_x + Tpartial_zA_y – Zpartial_xA_z,
-mu_0^{-1}nabla^2B_z &= T(partial_xA_x-partial_yA_x) + Ypartial_xA_y – Xpartial_yA_x,
end{split}
$$

where $X$, $Y$, $Z$ and $T$ are constant coefficients and we have assumed time-independence. These equations are a result of applying the electrostatic Maxwell’s equation $nablatimesmathbf{B} = mu_0mathbf{j}$
to the current density

$$
begin{split}
j_x &= XA_x+TA_y,
j_y &= TA_x+YA_y,
j_z &= ZA_z,
end{split}
$$
which was obtained by extremizing the free energy of an unconventional superconductor.

My question is how to translate this into a set of equations for the magnetic field component $B_i$ only, i.e., how to eliminate the vector potential components from these partial differential equations? Is it through some gauge choice in general possible to create a one to one relationship going between vector potential gradients and magnetic field components?

I’m following the notes on p. 43 of "Introduction to Unconventional Superconductivity", and according to Sigrist I should be able to put these equations on the form
$$
begin{split}
nabla^2B_x &= lambda_1^{-2}B_x + lambda_3^{-2}B_y
nabla^2B_y &= lambda_3^{-2}B_x + lambda_2^{-2}B_y
nabla^2B_z &= lambda_4^{-2}B_z.
end{split}
$$

---

What I’ve tried so far

Coulomb Gauge

Applying the Coulomb gauge we get $mu_0mathbf{j} = -nabla^2mathbf{A}$., which yields the set of equations
$$
begin{split}
nabla^2A_x &= -mu_0(XA_x+TA_y)
nabla^2A_y &= -mu_0(TA_x+YA_y)
nabla^2A_z &= -mu_0ZA_z,
end{split}
$$
when the current density above is inserted. This has exactly the same form as the equations of Sigrist, but for the vector potential components, and not the magnetic field ones. Could it be that Sigrist just mis-typed and meant to write this instead?

Axial gauge

Using the axial gauge, we can assume that $A_z=0$. This implies that $B_x = -partial_zA_y$ and $B_y = partial_zA_x$, by insertion in the equation $mathbf{B} = nablatimesmathbf{A}$. Inserting this above then yields the set of equations
$$
begin{split}
-mu_0^{-1}nabla^2B_x &= -TB_y + YB_x
-mu_0^{-1}nabla^2B_y &= XB_y – TB_x
-mu_0^{-1}nabla^2B_z &= T(partial_xA_x-partial_yA_y) + YB_z + (Y-X)partial_yA_x.
end{split}
$$
I thus get equations for $B_x$ and $B_y$ as desired, but with remaining vector potential components in the equations for $B_z$. Would it be possible to eliminate these with a residual gauge condition? Maybe I’ve misunderstood the axial gauge?