  We analyze the three-dimensional buckling of an elastic filament in a shear
flow of a viscous fluid at low Reynolds number and high Peclet number. We apply
the Euler-Bernoulli beam (elastica) theoretical model. We show the universal
character of the full 3D spectral problem for the small perturbation of the
thin filament from a straight position of arbitrary orientation. We use the
eigenvalues and eigenfunctions for the linearized elastica equation in the
shear plane, found earlier by [Liu et al., 2024] with the Chebyshev spectral
collocation method, to solve the full 3D eigenproblem. We provide a simple
analytic approximation to the eigenfunctions, represented as Gaussian
wavepackets. As the main result of the paper, we derive square-root dependence
of the eigenfunction wavenumber on the parameter $\tilde{\chi}=-\eta \sin 2\phi
\sin^2\theta$, where $\eta$ is the elastoviscous number, and the filament
orientation is determined by the zenith angle $\theta$ with respect to the
vorticity direction and the azimuthal angle $\phi$ relative to the flow
direction. We also compare the eigenfunctions with shapes of slightly buckled
elastic filaments with a non-negligible thickness with the same Young's
modulus, using the bead model and performing numerical simulations with the
precise Hydromultipole numerical codes.
