Integrals of motion in $WE_6$ CFT and the ODE/IM correspondence

We study the ODE/IM correspondence for the ordinary differential equation associated with the affine Lie algebra $E_6^{(1)}$. The WKB expansion of the solution of the ODE is performed by the diagonalization method, and the period integrals of the WKB coefficients along the Pochhammer contour are calculated. We also compute the integrals of motion on a cylinder in two-dimensional conformal field theory with W-symmetry associated with $E_6^{(1)}$. Their eigenvalues on the highest-weight state are shown to agree with the period integrals up to the sixth order.