Symmetry integrability and meromorphic extension

Evolution equations of the form $u_t=u_n+P(u_1,\dots,u_{n-1})$, where $n>1$ is a positive integer, $u(x,t)$ is the unknown function, $u_t$ is the partial derivative of $u$ with respect to $t$, $u_j$ is the $j$th partial derivative of $u$ with respect to $x$, and $P$ is a polynomial without constant and linear terms, split into equivalence classes with respect to the relation "be symmetries of each other". In this talk we describe all non-singleton equivalence classes of equations with weighted homogeneous right-hand side possessing the following meromorphic extension property: any local holomorphic solution $u(x,t)$ is a globally meromorphic function of $x$ for every fixed value of $t$.