Why Gaussian Beams are Eigensolution?

On the book principle of lasers by O. Svelto, it is written that Gaussian beams are eigensolutions of the paraxial wave equation (written in integral form).
I don’t understand why these beams should be eigensolutions of this equation, can you give me an explanation?

I don’t know if it can help but these are the equations of the book:

$vec E=u(x,y,z)e^{-ikz}e^{iomega t}$ is the em wave (paraxial approximation along z axis)

$begin {bmatrix} A&BC&Dend {bmatrix}$ is the matrix of an optical instrument

$S$ is the aperture of the optical element

$u(x,y,z)=frac 1 {B lambda}int_S u(x_1,y_1,z_1) exp(-ik frac {A(x_1^2+y_1^2)+D(x^2+y^2)-2x_1x-2y_1y} {2B}) dx_1 dy_1 $ is the paraxial wave equation in integral form

$u(x,y,z)=frac {u_0}{A+B/q_1} exp(-ik frac {x^2+y^2}{2q}) $ is the gaussian beam

where $q=frac {Aq_1+b}{Cq_1+D}$