[quote=olivant]I have never understood why Sollozzo would want to murder Vito if it was Vito's political and judicial help he was seeking. quote]

I think Sol's reasoning was that Vito put him in an inimical position: he couldn't do business without Vito's support, so for Sol it was either withdraw to Sicily with his tail twixt his legs--or try to kill Vito. His rational was as a "man of honor," plus he figured he could get the Corleones' remaining police/political protection at cut rates from the survivors.
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But I see Barzini egging him on. Barzini would never attempt to kill Vito on his own. But if Sol were willing to, what did he have to lose? If Sol succeeded, Barz would automatically become top guy. The judges and politicians who quit the Corleones would still have kids to put through college, and would be up for grabs--meaning Barzini's money he'd be making with Sol. If Sol failed, the Corleones would come after Tattaglia. And even if it came to war (as it did), the biggest guy always has the most to lose.