Is Lyme disease in New Zealand??

I really need to know if Lyme diesase is present in New Zealand because Luca might have shown some symptoms a few months ago. (sudden behaviour change, rash on neck, infected tick bite) but it might have been that stupid anti-insect shampoo the neighbours "secretly" washed him with that's another story, though...if you want, I can tell you. But for now, I just want to know if in New Zealand, a horse can get Lyme disease??

Generally, if ticks exist in your area Lyme disease is always a risk.
I did a quick Google search on New Zealand Lyme disease and there has been a few outbreaks in recent years.

Oh, because I heard that the horse can only get lymes disease through and infection of a certain type of tick...something like a black-legged ticked I heard that there are only cattle ticks here in NZ. Can horses get lymes disease from those??? I'm really worried....

About a month ago, Luca got a rash on his neck. Only on one side. It wasn't sore, bleeding or itchy, just small bumps on his neck, don't know how he got them. Then, he became aggressive, jumping at me with his teeth bared. I disciplined him, and now he doesn't bite anymore. But always when he comes up to me, he pins his ears back but NEVER bites or nips. Just makes grumpy faces. I heard that this is one of the Lymes diesase symptoms, but he doesn't do it with other people, just me. I think he's figuring out my place in the herd, and I'm working in showing him that I AM the leader. He has smooth, flowing actions but yesterday I noticed his trot was a bit stiff. Also, he sometimes puts back his ears and stomps his hind leg when I squeeze his sides or tap him with the whip. But he's just been broken in, so he's still learning commands and he's sometimes quite rebellious when he feels lazy. Other than that, he's completely healthy! But still....I'm worried..he's my first horse. What if he does have it???

Oh, and the neck rash wasn't a "bullseye" one. Just little bumps...but still..could it be?