I need to compute monodromy matrices $M(v)$, associated to a Frobenius structure for $A_n$ singularity with flat coordinates $v_1,\dots,v_n$, that is, $f(x)=x^{n+1}$. (due to Saito, Dubrovin etc.) Namely, i follow the paper by Dubrovin "WDVV Equations and Frobenius Manifolds" (2006). For $n=1$ the prepotential is $F(v)=v_1^3$, and is connected to Witten-Konstevich tau-function. Denote $g^{\alpha,\beta}=E^{\gamma}(v)c^{\alpha,\beta}_{\gamma}(v)$ with $E(v)$ an Euler vector field and $c^{\gamma}_{\alpha,\beta}(v)=\eta^{\gamma,\epsilon}\frac{\partial^3 F(v)}{\partial v_{\epsilon}\partial v_{\alpha} \partial v_{\beta}}$. However, I'm confused how to compute $M(v)$, since the only eigenvalue $u_1=\lambda=0$, which is a root of $\mathrm{det}(g^{\alpha,\beta}-\lambda\eta^{\alpha,\beta})$, is zero ($\eta$ is a triple derivative of $F$, and $E$ is a first order diff. operator). But to find $M$, as i see, one has to solve the following differential equation: $\frac{\partial}{\partial u_i}M=\{M,1/2\sum_{j\neq i}\frac{M_{i,j}^2}{u_i-u_j}\}$. Also, the same problem arises for $n=2$. Perhaps I miss something..

Thanks for any help!