It's known that there are sentences of first-order logic which only have infinite models, even if our language consists only of a binary relation $R$. An example of such a sentence is
$$\forall x \exists y Rxy \wedge \forall x \forall y \forall z ((Rxy \wedge Ryz ) \to Rxz) \wedge \neg \exists x Rxx$$
What I'm curious about is whether there are still sentences with only infinite models when we mandate that our binary relation be symmetric?
To phrase the question formally, consider a first-order language $\mathcal{L} = \{R^{2}\}$ (without equality). Let $\phi$ denote the formula
$$\forall x \forall y (Rxy \to Ryx)$$
Is there a sentence $\psi$ of $\mathcal{L}$ such that $\phi \wedge \psi$ has an infinite model, but no finite models?