Given a Diophantine equation it is not decidable if it has integer solution.

I. Is there a Diophantine set $\mathcal D_{unique}$ satisfying the properties

every member in $\mathcal D_{unique}$ is a polynomial

every member in $\mathcal D_{unique}$ has one integral point or less

every polynomial $f$ is reducible in finite time to a polynomial in $\mathcal D_{unique}$ satisfying the property $f$ has no integral solution iff the mapped polynomial has no integral solution?

II. Is there an universal set $\mathcal D_{unique}^{univeral}$ which models $\mathcal D_{unique}$?

Essentially given a diophantine equation, can we produce another with at most one integer solution, and which has an integer solution if and only if the first one does?

Assume $f$ is taken in $\mathbb Z[x_1,\dots,x_t]$ and has terms only of form $$b\prod_{i=1}^tx_i^{a_i}$$ where $a_i\in\{0,1\}$ and $b\in\mathbb Z$.

III. Perhaps in the above scenario an unique integral solution polynomial reducible situation is feasible.

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