Let \(u\) be a linear form acting on the vector space of polynomials with complex coefficients, then \((u)_n = \langle u, x^n \rangle\) are the moments of \(u\) and \(S(u)(z) = - \sum_{n=0}^\infty (u)_nz^{-n-1}\) is its formal Stieltjes transform. Laguerre-Hahn forms are such that \(S(u)\) satisfies the Riccati equation \(A(z)S'(u)(z) = B(z) S^2(u)(z) + C(z)S(u)(z) + D(z)\), where \(A,B,C,D\) are polynomials. There are infinitely many Riccati equations for a Laguerre-Hahn form, but the authors define the class of a Laguerre-Hahn form in terms of the ‘simplest’ possible equation. This class is zero or a positive integer. The Laguerre-Hahn forms of class zero have been described in a Ph.D. thesis of H. Bouakkaz (Paris, 1990). In the present paper the Laguerre-Hahn forms of class one with \((u)_{2n+1}=0\) for every integer \(n\) are completely described. It turns out that there are three canonical cases, namely \(A(x)=x\), \(A(x)=x^3\), and \(A(x)=x(x^2-1)\). Each case is considered. Some particular cases correspond to the forms associated with (generalized) Hermite polynomials, modified Lommel polynomials, symmetric Pollaczek polynomials, and Gegenbauer polynomials. A complete description of the obtained forms through suitable measures is not yet available.