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The phrase "a single solution but" is correct and usable in written English.
It can be used when introducing a contrast or exception to the idea of having only one solution.
Example: "There may be a single solution but it is not the only option worth considering."
Alternatives: "one solution, yet" or "a sole solution, however".
Exact(8)
Multi-objective optimization algorithms search for balanced solutions, providing not a single solution but a collection of them, an approximation to the Pareto set.
The solution of a multi-objective optimization problem is not a single solution but a complete non-dominated or Pareto set, which includes the alternatives representing potential compromise solutions among the objectives.
There follows an account of the problem of the instant of change, where it is concluded that the problem is too general to admit a single solution, but requires specification of further metaphysical principles envisaged as constraints on a type of solution.
By returning not just a single solution but a Pareto-optimal set of solutions (i.e., operation sequences, with appropriate part orientations and tools assigned) we have left the engineers freedom to apply further, not yet modeled parts of their domain knowledge.
It is unlikely that we will find a single solution, but increasing workplace flexibility is a step in the right direction.
A well-designed Forth program reads like natural language, and implements not just a single solution, but also sets of tools to attack related problems.
Similar(51)
But a single solution cannot capture all content.
The strongest conservative argument against abortion is that the Supreme Court overstepped the mark by imposing a single solution on a diverse country; but since then the Republicans have used their temporary dominance in Washington to trample on states' rights.Such overbearing behaviour is not a monopoly of the right.
The two problems have a single solution.
Each dilemma has a single solution.
Nor is there a single solution," Kember says.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com