Briefly I am wondering when a smooth foliation is a solution to a type of differential equation. I will link this related question which helped me form the following question.
Consider the space of smooth foliations $\mathcal H,$ of $M:=\Bbb R^2 - \lbrace 0 \rbrace$ with metric $g:=ds^2=dx^2+dy^2.$ With a given diffeomorphism $p:M\to M$ we can map each foliation $h_{\alpha}\in\mathcal{H}$ to the image space. Denote the foliations in the image space by $f_{\alpha}\in \mathcal{F}.$ Consider the $3$-tuple $(M,\mathcal{F},g).$ Consider $c(t)\varphi_{tt}=\pm h(x) \varphi_x.$ This is a heat equation.
Which diffeomorphism maximizes the number of solutions by $\mathcal{F}$ to the differential equation?
My attempt:
Prescribe the foliation $h_1=\lbrace \pm t/x: t>0 \rbrace$ on $M.$ Apply the diffeomorphism $p:M \to M$ with $p(x,y)=(e^x,e^y).$ Then obtain the foliation $f_1=\lbrace e^{\frac{\mp t}{\log x}}: t>0 \rbrace$ in the image space. Collect $(M,f_1,g)$ and plug in $f_1$ to the differential equation. See that it's satisfied when $c(t)=t$ and $h(x)=x.$ Therefore the lower bound on the number of solutions is $\ge1.$ I don't see how to maximize the number of solutions in general however.