Second-order evolution equations of divergent type for solenoidal vector fields on $\mathbb{R}^3$

In this paper, we describe the class $\mathfrak{K}_2^{(0)}(\mathbb{R}^3)$ of second-order differential operators of divergent type that are invariant under translations of $\mathbb{R}^3$ and are transformed covariantly under rotations of $\mathbb{R}^3$. Using such operators, one can construct evolutional equations that describe a translation-invariant dynamics of a solenoidal vector field $\boldsymbol{V}(\boldsymbol{x},t)$ so that each operator of the class $\mathfrak{K}_2^{(0)}(\mathbb{R}^3)$ determines an infinitesimal $t$-shift of this field. Also, we prove that the class of all evolutional equations for a unimodal vector field $\boldsymbol{V}(\boldsymbol{x},t)$ is trivial.