Is the Uniqueness of IVP Solutions Always Binary?

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Are these statements correct, if not could you give me an example

1. If solution of IVP is non-unique then there are infinitely many solutions
in short, if the solution to the IVP has at least 2 solutions then there are infinitely many solutions to this IVP

2.there are none IVP first order ODE's with finite solutions

for example there are no such IVP's that have only 2 or 3 solutions it must be either one or infinite

3. there is no first order linear ODE's that have more than 1 solution for each different initial conditions if the solution exists
 
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