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For each of the simple Lie algebras $\mathfrak{g}=A_l$, $D_l$ or $E_6$, weshow that the all-genera one-point FJRW invariants of $\mathfrak{g}$-type,after multiplication by suitable products of Pochhammer symbols, are thecoefficients of an algebraic generating function and hence are integral.Moreover, we find that the all-genera invariants themselves coincide with thecoefficients of the unique calibration of the Frobenius manifold of$\mathfrak{g}$-type evaluated at a special point. For the $A_4$ (5-spin) casewe also find two other normalizations of the sequence that are again integraland of at most exponential growth, and hence conjecturally are the Taylorcoefficients of some period functions.

Geometry and arithmetic of integrable hierarchies of KdV type. I. Integrality