They can, Most of the UK M&M have been described by a well known Icelandic trainer as trotty tölters. He watched the M&M classes as the Royal show one year and was facinated.
Charlie can tölt if he has a rider on him who knows how to ask for it. He is a Shetland, Fivla has been known to tölt when we have been asking her to do a really collected trot.
In the old days lots of horses ambled or had lateral gaits, they were very, very comfy to ride and ladies used them for side saddle, they were known as palfreys or amblers.
However it is an inefficient gait to have if you are pulling a load, you want a trotter, so when road became better and better and carts and carriages became more popular tölting was undesirable and nobody bred for the gait specifically. They bred for trtotting carriage horses.
In Iceland you cannot traverse the country by cart as the lave and ice fields change day by day, so they rode and used pack horses, so it was more important to have a comfy gaited horse to ride and carry a pack than it was to have a trotty cart horse, so for hundreds of years the Icelandic has been bred in isolation for the gaits.
? Can anyone with knowledge of gaited horses help me out 
